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Properties

Label 2-1332-37.2-c0-0-0
Degree 22
Conductor 13321332
Sign 0.873+0.487i0.873 + 0.487i
Analytic cond. 0.6647540.664754
Root an. cond. 0.8153240.815324
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 − 0.673i)7-s + (−0.939 − 1.34i)13-s + (−0.515 + 0.0451i)19-s + (0.642 + 0.766i)25-s + (−0.597 − 0.597i)31-s + (0.5 + 0.866i)37-s + (−1.40 + 1.40i)43-s + (2.20 − 1.85i)49-s + (1.15 − 0.811i)61-s + (0.524 + 1.43i)67-s − 1.53i·73-s + (0.0736 − 0.157i)79-s + (−2.64 − 1.85i)91-s + (0.515 + 1.92i)97-s + (−0.5 + 1.86i)103-s + ⋯
L(s)  = 1  + (1.85 − 0.673i)7-s + (−0.939 − 1.34i)13-s + (−0.515 + 0.0451i)19-s + (0.642 + 0.766i)25-s + (−0.597 − 0.597i)31-s + (0.5 + 0.866i)37-s + (−1.40 + 1.40i)43-s + (2.20 − 1.85i)49-s + (1.15 − 0.811i)61-s + (0.524 + 1.43i)67-s − 1.53i·73-s + (0.0736 − 0.157i)79-s + (−2.64 − 1.85i)91-s + (0.515 + 1.92i)97-s + (−0.5 + 1.86i)103-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((0.873+0.487i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=((0.873+0.487i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: 0.873+0.487i0.873 + 0.487i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(1297,⋅)\chi_{1332} (1297, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), 0.873+0.487i)(2,\ 1332,\ (\ :0),\ 0.873 + 0.487i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2444878131.244487813
L(12)L(\frac12) ≈\approx 1.2444878131.244487813
L(1)L(1) 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 1
3 1 1
37 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(−0.642−0.766i)T2 1 + (-0.642 - 0.766i)T^{2}
7 1+(−1.85+0.673i)T+(0.766−0.642i)T2 1 + (-1.85 + 0.673i)T + (0.766 - 0.642i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.939+1.34i)T+(−0.342+0.939i)T2 1 + (0.939 + 1.34i)T + (-0.342 + 0.939i)T^{2}
17 1+(0.342+0.939i)T2 1 + (0.342 + 0.939i)T^{2}
19 1+(0.515−0.0451i)T+(0.984−0.173i)T2 1 + (0.515 - 0.0451i)T + (0.984 - 0.173i)T^{2}
23 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
29 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
31 1+(0.597+0.597i)T+iT2 1 + (0.597 + 0.597i)T + iT^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(1.40−1.40i)T−iT2 1 + (1.40 - 1.40i)T - iT^{2}
47 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
53 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
59 1+(0.642−0.766i)T2 1 + (0.642 - 0.766i)T^{2}
61 1+(−1.15+0.811i)T+(0.342−0.939i)T2 1 + (-1.15 + 0.811i)T + (0.342 - 0.939i)T^{2}
67 1+(−0.524−1.43i)T+(−0.766+0.642i)T2 1 + (-0.524 - 1.43i)T + (-0.766 + 0.642i)T^{2}
71 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
73 1+1.53iT−T2 1 + 1.53iT - T^{2}
79 1+(−0.0736+0.157i)T+(−0.642−0.766i)T2 1 + (-0.0736 + 0.157i)T + (-0.642 - 0.766i)T^{2}
83 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
89 1+(−0.642+0.766i)T2 1 + (-0.642 + 0.766i)T^{2}
97 1+(−0.515−1.92i)T+(−0.866+0.5i)T2 1 + (-0.515 - 1.92i)T + (-0.866 + 0.5i)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

−9.926324758383409116539055704698, −8.795305771154110168948889508193, −7.86899562245977610530765631671, −7.69125641207217360742208235392, −6.55025601217139659162493751853, −5.20155837222902189536215190991, −4.91368697205723218659000058226, −3.77401048436069741270479138675, −2.47150732919114263874680433570, −1.24084188896274624280642742284, 1.72905186129401859337706077276, 2.43410113858641946287121224221, 4.11061941034522947012146270662, 4.83321656062107793627133931057, 5.50714778317462828764922993450, 6.73430463999861716760983550690, 7.46100969658141556416640213980, 8.470037252276384233910469305741, 8.822699257324894203508307389907, 9.847936957115270335439062319507

Graph of the ZZ-function along the critical line