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Properties

Label 2-1014-1.1-c3-0-24
Degree 22
Conductor 10141014
Sign 11
Analytic cond. 59.827959.8279
Root an. cond. 7.734857.73485
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 10·5-s + 6·6-s + 8·7-s + 8·8-s + 9·9-s − 20·10-s − 40·11-s + 12·12-s + 16·14-s − 30·15-s + 16·16-s + 130·17-s + 18·18-s + 20·19-s − 40·20-s + 24·21-s − 80·22-s + 24·24-s − 25·25-s + 27·27-s + 32·28-s − 18·29-s − 60·30-s + 184·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.431·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.09·11-s + 0.288·12-s + 0.305·14-s − 0.516·15-s + 1/4·16-s + 1.85·17-s + 0.235·18-s + 0.241·19-s − 0.447·20-s + 0.249·21-s − 0.775·22-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.215·28-s − 0.115·29-s − 0.365·30-s + 1.06·31-s + ⋯

Functional equation

Λ(s)=(1014s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1014s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10141014    =    2⋅3⋅1322 \cdot 3 \cdot 13^{2}
Sign: 11
Analytic conductor: 59.827959.8279
Root analytic conductor: 7.734857.73485
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1014, ( :3/2), 1)(2,\ 1014,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.7588577803.758857780
L(12)L(\frac12) ≈\approx 3.7588577803.758857780
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)
bad2 1−pT 1 - p T
3 1−pT 1 - p T
13 1 1
good5 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
7 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
11 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
17 1−130T+p3T2 1 - 130 T + p^{3} T^{2}
19 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
31 1−184T+p3T2 1 - 184 T + p^{3} T^{2}
37 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
41 1−362T+p3T2 1 - 362 T + p^{3} T^{2}
43 1−76T+p3T2 1 - 76 T + p^{3} T^{2}
47 1−452T+p3T2 1 - 452 T + p^{3} T^{2}
53 1−382T+p3T2 1 - 382 T + p^{3} T^{2}
59 1+464T+p3T2 1 + 464 T + p^{3} T^{2}
61 1−358T+p3T2 1 - 358 T + p^{3} T^{2}
67 1−700T+p3T2 1 - 700 T + p^{3} T^{2}
71 1−748T+p3T2 1 - 748 T + p^{3} T^{2}
73 1+1058T+p3T2 1 + 1058 T + p^{3} T^{2}
79 1+976T+p3T2 1 + 976 T + p^{3} T^{2}
83 1−1008T+p3T2 1 - 1008 T + p^{3} T^{2}
89 1−386T+p3T2 1 - 386 T + p^{3} T^{2}
97 1−614T+p3T2 1 - 614 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

−9.703974263994165601957947360196, −8.435113713446498097304087181019, −7.72330190374658220047622912186, −7.41812313424302815665877894826, −5.96954528153293445114661133586, −5.12485242198336402734946153796, −4.18184017096352409463594465195, −3.31826065918618045217441523449, −2.43039462383601003556197025285, −0.925280667783231477268077905844, 0.925280667783231477268077905844, 2.43039462383601003556197025285, 3.31826065918618045217441523449, 4.18184017096352409463594465195, 5.12485242198336402734946153796, 5.96954528153293445114661133586, 7.41812313424302815665877894826, 7.72330190374658220047622912186, 8.435113713446498097304087181019, 9.703974263994165601957947360196

Graph of the ZZ-function along the critical line