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Properties

Label 2-1332-37.18-c0-0-0
Degree 22
Conductor 13321332
Sign −0.566+0.823i-0.566 + 0.823i
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 − 0.657i)13-s + (0.168 − 1.92i)19-s + (−0.642 + 0.766i)25-s + (−1.28 − 1.28i)31-s + (0.5 − 0.866i)37-s + (−0.123 + 0.123i)43-s + (2.20 + 1.85i)49-s + (−0.811 + 1.15i)61-s + (−0.524 + 1.43i)67-s − 1.53i·73-s + (1.80 − 0.842i)79-s + (1.29 + 1.85i)91-s + (−0.168 − 0.0451i)97-s + (−0.5 + 0.133i)103-s + ⋯
L(s)  = 1  + (−1.85 − 0.673i)7-s + (−0.939 − 0.657i)13-s + (0.168 − 1.92i)19-s + (−0.642 + 0.766i)25-s + (−1.28 − 1.28i)31-s + (0.5 − 0.866i)37-s + (−0.123 + 0.123i)43-s + (2.20 + 1.85i)49-s + (−0.811 + 1.15i)61-s + (−0.524 + 1.43i)67-s − 1.53i·73-s + (1.80 − 0.842i)79-s + (1.29 + 1.85i)91-s + (−0.168 − 0.0451i)97-s + (−0.5 + 0.133i)103-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((−0.566+0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s) L(s)=((−0.566+0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: −0.566+0.823i-0.566 + 0.823i
Analytic conductor: 0.6647540.664754
Root analytic conductor: 0.8153240.815324
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1332(721,⋅)\chi_{1332} (721, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :0), −0.566+0.823i)(2,\ 1332,\ (\ :0),\ -0.566 + 0.823i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.50498735510.5049873551
L(12)L(\frac12) ≈\approx 0.50498735510.5049873551
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+0.657i)T+(0.342+0.939i)T2 1 + (0.939 + 0.657i)T + (0.342 + 0.939i)T^{2}
17 1+(−0.342+0.939i)T2 1 + (-0.342 + 0.939i)T^{2}
19 1+(−0.168+1.92i)T+(−0.984−0.173i)T2 1 + (-0.168 + 1.92i)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+(1.28+1.28i)T+iT2 1 + (1.28 + 1.28i)T + iT^{2}
41 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
43 1+(0.123−0.123i)T−iT2 1 + (0.123 - 0.123i)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+(0.811−1.15i)T+(−0.342−0.939i)T2 1 + (0.811 - 1.15i)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+(−1.80+0.842i)T+(0.642−0.766i)T2 1 + (-1.80 + 0.842i)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+(0.168+0.0451i)T+(0.866+0.5i)T2 1 + (0.168 + 0.0451i)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

−9.392260685012751961610570287090, −9.170337697250141729081520463292, −7.55325936776091187030723233446, −7.23561438585920548687584866610, −6.29262214612466102001980359763, −5.44811283756963351702039574318, −4.28573659152661014041654191763, −3.32841181680964320142224985181, −2.51177254433627628489867784321, −0.39427750236906976356026398143, 2.00522407425586386852916414988, 3.12801713952591114319004930112, 3.89485036267912577302139568733, 5.20577793816332326247710498479, 6.10504159341156488054598197598, 6.66089459558415994355222556242, 7.61995113886796291216933857644, 8.598937055363214273633810543706, 9.549481777389093294177599514786, 9.812068523232340843838557427239

Graph of the ZZ-function along the critical line