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Properties

Label 2-396-11.4-c1-0-1
Degree 22
Conductor 396396
Sign 0.350−0.936i0.350 - 0.936i
Analytic cond. 3.162073.16207
Root an. cond. 1.778221.77822
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 + 0.951i)5-s + (0.0729 + 0.224i)7-s + (0.809 + 3.21i)11-s + (3.42 + 2.48i)13-s + (−0.118 + 0.0857i)17-s + (−2.11 + 6.51i)19-s + 5·23-s + (−0.736 + 2.26i)25-s + (−0.618 − 1.90i)29-s + (−2.73 − 1.98i)31-s + (−0.309 − 0.224i)35-s + (0.690 + 2.12i)37-s + (2.69 − 8.28i)41-s − 2.52·43-s + (−2.95 + 9.09i)47-s + ⋯
L(s)  = 1  + (−0.585 + 0.425i)5-s + (0.0275 + 0.0848i)7-s + (0.243 + 0.969i)11-s + (0.950 + 0.690i)13-s + (−0.0286 + 0.0207i)17-s + (−0.485 + 1.49i)19-s + 1.04·23-s + (−0.147 + 0.453i)25-s + (−0.114 − 0.353i)29-s + (−0.491 − 0.357i)31-s + (−0.0522 − 0.0379i)35-s + (0.113 + 0.349i)37-s + (0.420 − 1.29i)41-s − 0.385·43-s + (−0.431 + 1.32i)47-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=((0.350−0.936i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+1/2) L(s)=((0.350−0.936i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 0.350−0.936i0.350 - 0.936i
Analytic conductor: 3.162073.16207
Root analytic conductor: 1.778221.77822
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ396(37,⋅)\chi_{396} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 396, ( :1/2), 0.350−0.936i)(2,\ 396,\ (\ :1/2),\ 0.350 - 0.936i)

Particular Values

L(1)L(1) ≈\approx 0.974536+0.675816i0.974536 + 0.675816i
L(12)L(\frac12) ≈\approx 0.974536+0.675816i0.974536 + 0.675816i
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}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
11 1+(−0.809−3.21i)T 1 + (-0.809 - 3.21i)T
good5 1+(1.30−0.951i)T+(1.54−4.75i)T2 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2}
7 1+(−0.0729−0.224i)T+(−5.66+4.11i)T2 1 + (-0.0729 - 0.224i)T + (-5.66 + 4.11i)T^{2}
13 1+(−3.42−2.48i)T+(4.01+12.3i)T2 1 + (-3.42 - 2.48i)T + (4.01 + 12.3i)T^{2}
17 1+(0.118−0.0857i)T+(5.25−16.1i)T2 1 + (0.118 - 0.0857i)T + (5.25 - 16.1i)T^{2}
19 1+(2.11−6.51i)T+(−15.3−11.1i)T2 1 + (2.11 - 6.51i)T + (-15.3 - 11.1i)T^{2}
23 1−5T+23T2 1 - 5T + 23T^{2}
29 1+(0.618+1.90i)T+(−23.4+17.0i)T2 1 + (0.618 + 1.90i)T + (-23.4 + 17.0i)T^{2}
31 1+(2.73+1.98i)T+(9.57+29.4i)T2 1 + (2.73 + 1.98i)T + (9.57 + 29.4i)T^{2}
37 1+(−0.690−2.12i)T+(−29.9+21.7i)T2 1 + (-0.690 - 2.12i)T + (-29.9 + 21.7i)T^{2}
41 1+(−2.69+8.28i)T+(−33.1−24.0i)T2 1 + (-2.69 + 8.28i)T + (-33.1 - 24.0i)T^{2}
43 1+2.52T+43T2 1 + 2.52T + 43T^{2}
47 1+(2.95−9.09i)T+(−38.0−27.6i)T2 1 + (2.95 - 9.09i)T + (-38.0 - 27.6i)T^{2}
53 1+(−5.35−3.88i)T+(16.3+50.4i)T2 1 + (-5.35 - 3.88i)T + (16.3 + 50.4i)T^{2}
59 1+(3.64+11.2i)T+(−47.7+34.6i)T2 1 + (3.64 + 11.2i)T + (-47.7 + 34.6i)T^{2}
61 1+(−8.78+6.37i)T+(18.8−58.0i)T2 1 + (-8.78 + 6.37i)T + (18.8 - 58.0i)T^{2}
67 1+4.38T+67T2 1 + 4.38T + 67T^{2}
71 1+(11.7−8.55i)T+(21.9−67.5i)T2 1 + (11.7 - 8.55i)T + (21.9 - 67.5i)T^{2}
73 1+(4.85+14.9i)T+(−59.0+42.9i)T2 1 + (4.85 + 14.9i)T + (-59.0 + 42.9i)T^{2}
79 1+(−7.04−5.11i)T+(24.4+75.1i)T2 1 + (-7.04 - 5.11i)T + (24.4 + 75.1i)T^{2}
83 1+(1.42−1.03i)T+(25.6−78.9i)T2 1 + (1.42 - 1.03i)T + (25.6 - 78.9i)T^{2}
89 1−5.18T+89T2 1 - 5.18T + 89T^{2}
97 1+(12.1+8.83i)T+(29.9+92.2i)T2 1 + (12.1 + 8.83i)T + (29.9 + 92.2i)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

−11.42779751903833990524404918715, −10.68124137215923633698081461041, −9.644119042258777059322869283600, −8.715199407439125973517940901249, −7.67347735837214704393782417591, −6.84844712028637821353997263443, −5.80361438710434876042063661569, −4.37647637538400691206784596825, −3.50423525813854271388238601978, −1.80853942679714421666427884745, 0.842577288317813414605117291850, 2.95794669535776452074243716417, 4.08871117031297673956140307819, 5.24666660033984875796236567599, 6.35431753915924871455654508631, 7.42930384808506014445648346910, 8.600570512594095172804076570627, 8.907206882097210255747216555563, 10.41798327317094666871518986704, 11.15384611041681204722246047447

Graph of the ZZ-function along the critical line