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Properties

Label 2-309-309.167-c0-0-0
Degree $2$
Conductor $309$
Sign $0.642 + 0.766i$
Analytic cond. $0.154211$
Root an. cond. $0.392697$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.982 + 0.183i)3-s + (0.739 − 0.673i)4-s + (−0.876 − 1.75i)7-s + (0.932 − 0.361i)9-s + (−0.602 + 0.798i)12-s + (0.658 + 1.32i)13-s + (0.0922 − 0.995i)16-s + (0.538 + 0.100i)19-s + (1.18 + 1.56i)21-s + (−0.850 − 0.526i)25-s + (−0.850 + 0.526i)27-s + (−1.83 − 0.710i)28-s + (−0.111 + 1.20i)31-s + (0.445 − 0.895i)36-s + (−0.537 + 0.711i)37-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)3-s + (0.739 − 0.673i)4-s + (−0.876 − 1.75i)7-s + (0.932 − 0.361i)9-s + (−0.602 + 0.798i)12-s + (0.658 + 1.32i)13-s + (0.0922 − 0.995i)16-s + (0.538 + 0.100i)19-s + (1.18 + 1.56i)21-s + (−0.850 − 0.526i)25-s + (−0.850 + 0.526i)27-s + (−1.83 − 0.710i)28-s + (−0.111 + 1.20i)31-s + (0.445 − 0.895i)36-s + (−0.537 + 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6527360066\)
\(L(\frac12)\) \(\approx\) \(0.6527360066\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.982 - 0.183i)T \)
103 \( 1 + (0.602 - 0.798i)T \)
good2 \( 1 + (-0.739 + 0.673i)T^{2} \)
5 \( 1 + (0.850 + 0.526i)T^{2} \)
7 \( 1 + (0.876 + 1.75i)T + (-0.602 + 0.798i)T^{2} \)
11 \( 1 + (-0.739 + 0.673i)T^{2} \)
13 \( 1 + (-0.658 - 1.32i)T + (-0.602 + 0.798i)T^{2} \)
17 \( 1 + (-0.445 - 0.895i)T^{2} \)
19 \( 1 + (-0.538 - 0.100i)T + (0.932 + 0.361i)T^{2} \)
23 \( 1 + (-0.739 - 0.673i)T^{2} \)
29 \( 1 + (0.850 + 0.526i)T^{2} \)
31 \( 1 + (0.111 - 1.20i)T + (-0.982 - 0.183i)T^{2} \)
37 \( 1 + (0.537 - 0.711i)T + (-0.273 - 0.961i)T^{2} \)
41 \( 1 + (0.850 - 0.526i)T^{2} \)
43 \( 1 + (-1.02 - 1.35i)T + (-0.273 + 0.961i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.932 - 0.361i)T^{2} \)
59 \( 1 + (0.602 + 0.798i)T^{2} \)
61 \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \)
67 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
71 \( 1 + (0.850 - 0.526i)T^{2} \)
73 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
79 \( 1 + (0.243 - 0.857i)T + (-0.850 - 0.526i)T^{2} \)
83 \( 1 + (0.602 - 0.798i)T^{2} \)
89 \( 1 + (-0.0922 - 0.995i)T^{2} \)
97 \( 1 + (-1.44 + 0.895i)T + (0.445 - 0.895i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.51681524721732762161117173811, −10.85957664525650653306845386004, −10.12914428895467493610928154178, −9.460799673304488691631842931966, −7.50516487887005785325679978737, −6.69020253633633425579078356808, −6.18335867114909723218627050021, −4.71301008518075433565194025353, −3.62898575853660842174255863207, −1.30049452879220619474988422114, 2.32506086333377622741730086950, 3.57728933578770016329019917682, 5.61141753265880979232118929251, 5.90797008390197426625116375172, 7.10543591083731645883960277442, 8.134725305935256304220646476431, 9.258294141155660654337814370873, 10.37930040367435353343384624222, 11.39650784249685318237350542462, 12.00934465976821835938444142689

Graph of the $Z$-function along the critical line