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Properties

Label 4-7644e2-1.1-c1e2-0-7
Degree 44
Conductor 5843073658430736
Sign 11
Analytic cond. 3725.593725.59
Root an. cond. 7.812657.81265
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 3·9-s + 4·11-s + 2·13-s + 2·15-s − 7·19-s + 3·23-s − 25-s − 4·27-s − 3·29-s + 31-s − 8·33-s + 2·37-s − 4·39-s − 6·41-s + 5·43-s − 3·45-s + 7·47-s + 5·53-s − 4·55-s + 14·57-s + 16·59-s − 12·61-s − 2·65-s + 14·67-s − 6·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.60·19-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.557·29-s + 0.179·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.762·43-s − 0.447·45-s + 1.02·47-s + 0.686·53-s − 0.539·55-s + 1.85·57-s + 2.08·59-s − 1.53·61-s − 0.248·65-s + 1.71·67-s − 0.722·69-s + ⋯

Functional equation

Λ(s)=(58430736s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(58430736s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5843073658430736    =    24⋅32⋅74⋅1322^{4} \cdot 3^{2} \cdot 7^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 3725.593725.59
Root analytic conductor: 7.812657.81265
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 58430736, ( :1/2,1/2), 1)(4,\ 58430736,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8944187961.894418796
L(12)L(\frac12) ≈\approx 1.8944187961.894418796
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C1C_1 (1+T)2 ( 1 + T )^{2}
7 1 1
13C1C_1 (1−T)2 ( 1 - T )^{2}
good5D4D_{4} 1+T+2T2+pT3+p2T4 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4}
11C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19D4D_{4} 1+7T+42T2+7pT3+p2T4 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4}
23D4D_{4} 1−3T+40T2−3pT3+p2T4 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4}
29D4D_{4} 1+3T+52T2+3pT3+p2T4 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−T+54T2−pT3+p2T4 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4}
37D4D_{4} 1−2T+42T2−2pT3+p2T4 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+6T+58T2+6pT3+p2T4 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−5T+18T2−5pT3+p2T4 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−7T+98T2−7pT3+p2T4 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−5T+104T2−5pT3+p2T4 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4}
59C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
61C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
67D4D_{4} 1−14T+150T2−14pT3+p2T4 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+6T+118T2+6pT3+p2T4 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+T−60T2+pT3+p2T4 1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4}
79D4D_{4} 1−5T+90T2−5pT3+p2T4 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4}
83D4D_{4} 1+5T+98T2+5pT3+p2T4 1 + 5 T + 98 T^{2} + 5 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−9T+190T2−9pT3+p2T4 1 - 9 T + 190 T^{2} - 9 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+29T+396T2+29pT3+p2T4 1 + 29 T + 396 T^{2} + 29 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.83422983549892817206374131692, −7.80646403650249034745819303763, −7.17339974202224055847078632513, −6.91374354999532100307991331989, −6.58621968710743022131789556477, −6.45196946005752336279303381949, −5.83439717668421762673464716387, −5.80124821209625027960061926441, −5.25697696782888812571226921188, −4.93447468015548704998185027680, −4.33948800390894330558888376112, −4.23794332031473584731802141805, −3.74456166261019688731733419535, −3.67961576420719775120124934790, −2.87873491698267737217202065508, −2.45659779218722649991064154283, −1.79461396008775756194560689391, −1.54989730858972343540937227116, −0.76092106459037558910641398196, −0.50785677616995654619015389337, 0.50785677616995654619015389337, 0.76092106459037558910641398196, 1.54989730858972343540937227116, 1.79461396008775756194560689391, 2.45659779218722649991064154283, 2.87873491698267737217202065508, 3.67961576420719775120124934790, 3.74456166261019688731733419535, 4.23794332031473584731802141805, 4.33948800390894330558888376112, 4.93447468015548704998185027680, 5.25697696782888812571226921188, 5.80124821209625027960061926441, 5.83439717668421762673464716387, 6.45196946005752336279303381949, 6.58621968710743022131789556477, 6.91374354999532100307991331989, 7.17339974202224055847078632513, 7.80646403650249034745819303763, 7.83422983549892817206374131692

Graph of the ZZ-function along the critical line