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Properties

Label 2-39e2-1.1-c3-0-182
Degree 22
Conductor 15211521
Sign −1-1
Analytic cond. 89.741989.7419
Root an. cond. 9.473229.47322
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s + 17·5-s − 20·7-s + 68·10-s − 32·11-s − 80·14-s − 64·16-s + 13·17-s − 30·19-s + 136·20-s − 128·22-s − 78·23-s + 164·25-s − 160·28-s − 197·29-s + 74·31-s − 256·32-s + 52·34-s − 340·35-s + 227·37-s − 120·38-s − 165·41-s − 156·43-s − 256·44-s − 312·46-s − 162·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.52·5-s − 1.07·7-s + 2.15·10-s − 0.877·11-s − 1.52·14-s − 16-s + 0.185·17-s − 0.362·19-s + 1.52·20-s − 1.24·22-s − 0.707·23-s + 1.31·25-s − 1.07·28-s − 1.26·29-s + 0.428·31-s − 1.41·32-s + 0.262·34-s − 1.64·35-s + 1.00·37-s − 0.512·38-s − 0.628·41-s − 0.553·43-s − 0.877·44-s − 1.00·46-s − 0.502·47-s + ⋯

Functional equation

Λ(s)=(1521s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1521s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15211521    =    32⋅1323^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 89.741989.7419
Root analytic conductor: 9.473229.47322
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1521, ( :3/2), −1)(2,\ 1521,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
13 1 1
good2 1−p2T+p3T2 1 - p^{2} T + p^{3} T^{2}
5 1−17T+p3T2 1 - 17 T + p^{3} T^{2}
7 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
11 1+32T+p3T2 1 + 32 T + p^{3} T^{2}
17 1−13T+p3T2 1 - 13 T + p^{3} T^{2}
19 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
23 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
29 1+197T+p3T2 1 + 197 T + p^{3} T^{2}
31 1−74T+p3T2 1 - 74 T + p^{3} T^{2}
37 1−227T+p3T2 1 - 227 T + p^{3} T^{2}
41 1+165T+p3T2 1 + 165 T + p^{3} T^{2}
43 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
47 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
53 1+93T+p3T2 1 + 93 T + p^{3} T^{2}
59 1+864T+p3T2 1 + 864 T + p^{3} T^{2}
61 1−145T+p3T2 1 - 145 T + p^{3} T^{2}
67 1+862T+p3T2 1 + 862 T + p^{3} T^{2}
71 1−654T+p3T2 1 - 654 T + p^{3} T^{2}
73 1+215T+p3T2 1 + 215 T + p^{3} T^{2}
79 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
83 1−628T+p3T2 1 - 628 T + p^{3} T^{2}
89 1+266T+p3T2 1 + 266 T + p^{3} T^{2}
97 1+238T+p3T2 1 + 238 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.920136294650283049030384821464, −7.66162194189476310668250123646, −6.49430710776968043618448412964, −6.11280905173866511994350357653, −5.44826736949723099989523146247, −4.64009483917479849855926390300, −3.47599737417239054431810970550, −2.72190868951251984021587235607, −1.87110376828255682208425364792, 0, 1.87110376828255682208425364792, 2.72190868951251984021587235607, 3.47599737417239054431810970550, 4.64009483917479849855926390300, 5.44826736949723099989523146247, 6.11280905173866511994350357653, 6.49430710776968043618448412964, 7.66162194189476310668250123646, 8.920136294650283049030384821464

Graph of the ZZ-function along the critical line