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Properties

Label 16-637e8-1.1-c1e8-0-6
Degree 1616
Conductor 2.711×10222.711\times 10^{22}
Sign 11
Analytic cond. 448056.448056.
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·8-s − 9-s + 2·11-s − 3·16-s + 2·18-s − 4·22-s − 12·23-s − 14·25-s + 2·29-s + 12·32-s − 3·36-s − 20·37-s + 14·43-s + 6·44-s + 24·46-s + 28·50-s + 48·53-s − 4·58-s − 2·64-s − 4·67-s − 16·71-s − 2·72-s + 40·74-s − 24·79-s + 15·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.707·8-s − 1/3·9-s + 0.603·11-s − 3/4·16-s + 0.471·18-s − 0.852·22-s − 2.50·23-s − 2.79·25-s + 0.371·29-s + 2.12·32-s − 1/2·36-s − 3.28·37-s + 2.13·43-s + 0.904·44-s + 3.53·46-s + 3.95·50-s + 6.59·53-s − 0.525·58-s − 1/4·64-s − 0.488·67-s − 1.89·71-s − 0.235·72-s + 4.64·74-s − 2.70·79-s + 5/3·81-s + ⋯

Functional equation

Λ(s)=((716⋅138)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((716⋅138)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 716⋅1387^{16} \cdot 13^{8}
Sign: 11
Analytic conductor: 448056.448056.
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 716⋅138, ( :[1/2]8), 1)(16,\ 7^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.3289026231.328902623
L(12)L(\frac12) ≈\approx 1.3289026231.328902623
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1+pT4+p4T8 1 + p T^{4} + p^{4} T^{8}
good2 (1+T−3T3−5T4−3pT5+p3T7+p4T8)2 ( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} )^{2}
3 1+T2−14T4−pT6+5p3T8−p3T10−14p4T12+p6T14+p8T16 1 + T^{2} - 14 T^{4} - p T^{6} + 5 p^{3} T^{8} - p^{3} T^{10} - 14 p^{4} T^{12} + p^{6} T^{14} + p^{8} T^{16}
5 (1+7T2+59T4+7p2T6+p4T8)2 ( 1 + 7 T^{2} + 59 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} )^{2}
11 (1−T+8T2+29T3−83T4+29pT5+8p2T6−p3T7+p4T8)2 ( 1 - T + 8 T^{2} + 29 T^{3} - 83 T^{4} + 29 p T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
17 1−16T2+251T4+9168T6−157480T8+9168p2T10+251p4T12−16p6T14+p8T16 1 - 16 T^{2} + 251 T^{4} + 9168 T^{6} - 157480 T^{8} + 9168 p^{2} T^{10} + 251 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16}
19 1−63T2+2258T4−62307T6+1364391T8−62307p2T10+2258p4T12−63p6T14+p8T16 1 - 63 T^{2} + 2258 T^{4} - 62307 T^{6} + 1364391 T^{8} - 62307 p^{2} T^{10} + 2258 p^{4} T^{12} - 63 p^{6} T^{14} + p^{8} T^{16}
23 (1+6T−6T2−24T3+407T4−24pT5−6p2T6+6p3T7+p4T8)2 ( 1 + 6 T - 6 T^{2} - 24 T^{3} + 407 T^{4} - 24 p T^{5} - 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2}
29 (1−T−54T2+3T3+2155T4+3pT5−54p2T6−p3T7+p4T8)2 ( 1 - T - 54 T^{2} + 3 T^{3} + 2155 T^{4} + 3 p T^{5} - 54 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2}
31 (1+72T2+2581T4+72p2T6+p4T8)2 ( 1 + 72 T^{2} + 2581 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} )^{2}
37 (1+10T+14T2+120T3+2327T4+120pT5+14p2T6+10p3T7+p4T8)2 ( 1 + 10 T + 14 T^{2} + 120 T^{3} + 2327 T^{4} + 120 p T^{5} + 14 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2}
41 1−8T2−2846T4+3616T6+5534755T8+3616p2T10−2846p4T12−8p6T14+p8T16 1 - 8 T^{2} - 2846 T^{4} + 3616 T^{6} + 5534755 T^{8} + 3616 p^{2} T^{10} - 2846 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16}
43 (1−7T+32T2+483T3−3667T4+483pT5+32p2T6−7p3T7+p4T8)2 ( 1 - 7 T + 32 T^{2} + 483 T^{3} - 3667 T^{4} + 483 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2}
47 (1+32T2−1059T4+32p2T6+p4T8)2 ( 1 + 32 T^{2} - 1059 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2}
53 (1−12T+129T2−12pT3+p2T4)4 ( 1 - 12 T + 129 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4}
59 1−184T2+18755T4−1497576T6+98070104T8−1497576p2T10+18755p4T12−184p6T14+p8T16 1 - 184 T^{2} + 18755 T^{4} - 1497576 T^{6} + 98070104 T^{8} - 1497576 p^{2} T^{10} + 18755 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16}
61 1−192T2+20258T4−1759488T6+124742451T8−1759488p2T10+20258p4T12−192p6T14+p8T16 1 - 192 T^{2} + 20258 T^{4} - 1759488 T^{6} + 124742451 T^{8} - 1759488 p^{2} T^{10} + 20258 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16}
67 (1+T−66T2+pT3+p2T4)4 ( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} )^{4}
71 (1+4T−55T2+4pT3+p2T4)4 ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4}
73 (1+240T2+25006T4+240p2T6+p4T8)2 ( 1 + 240 T^{2} + 25006 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} )^{2}
79 (1+6T+154T2+6pT3+p2T4)4 ( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4}
83 (1+280T2+33053T4+280p2T6+p4T8)2 ( 1 + 280 T^{2} + 33053 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} )^{2}
89 1−239T2+27262T4−3350063T6+376362199T8−3350063p2T10+27262p4T12−239p6T14+p8T16 1 - 239 T^{2} + 27262 T^{4} - 3350063 T^{6} + 376362199 T^{8} - 3350063 p^{2} T^{10} + 27262 p^{4} T^{12} - 239 p^{6} T^{14} + p^{8} T^{16}
97 1−141T2+74T4−139449T6+116727639T8−139449p2T10+74p4T12−141p6T14+p8T16 1 - 141 T^{2} + 74 T^{4} - 139449 T^{6} + 116727639 T^{8} - 139449 p^{2} T^{10} + 74 p^{4} T^{12} - 141 p^{6} T^{14} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.56728148578927855568062542163, −4.43859017682249900036757608598, −4.31366968624434947174578385894, −4.07281619270835394406863968169, −4.02865999138294273477824972901, −3.89277214277907347779548060582, −3.88842867203087388304476290008, −3.68960547837397041636009258577, −3.43497354338087349875385977757, −3.41924764994475015402629261393, −3.25949850134240769253431019742, −3.03620462983589682119790021823, −2.63077672105226735147720685904, −2.42441246894867687522291625691, −2.37829516638545847841447354908, −2.34319591011078898335992378749, −2.27767983567120718258852681783, −2.05412434316914290697350386237, −1.56284211725174995451881222743, −1.55673566926245904266820283336, −1.42916608395552063693088818392, −1.29433465208163791353622359503, −0.950068680988019740981576199563, −0.60362787116950152001889747997, −0.24180487227083459192103064091, 0.24180487227083459192103064091, 0.60362787116950152001889747997, 0.950068680988019740981576199563, 1.29433465208163791353622359503, 1.42916608395552063693088818392, 1.55673566926245904266820283336, 1.56284211725174995451881222743, 2.05412434316914290697350386237, 2.27767983567120718258852681783, 2.34319591011078898335992378749, 2.37829516638545847841447354908, 2.42441246894867687522291625691, 2.63077672105226735147720685904, 3.03620462983589682119790021823, 3.25949850134240769253431019742, 3.41924764994475015402629261393, 3.43497354338087349875385977757, 3.68960547837397041636009258577, 3.88842867203087388304476290008, 3.89277214277907347779548060582, 4.02865999138294273477824972901, 4.07281619270835394406863968169, 4.31366968624434947174578385894, 4.43859017682249900036757608598, 4.56728148578927855568062542163

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.