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Properties

Label 2-309-309.278-c0-0-0
Degree $2$
Conductor $309$
Sign $-0.640 + 0.768i$
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  + (−0.602 − 0.798i)3-s + (−0.850 − 0.526i)4-s + (−0.890 − 0.811i)7-s + (−0.273 + 0.961i)9-s + (0.0922 + 0.995i)12-s + (−1.25 − 1.14i)13-s + (0.445 + 0.895i)16-s + (1.18 − 1.56i)19-s + (−0.111 + 1.20i)21-s + (0.932 + 0.361i)25-s + (0.932 − 0.361i)27-s + (0.329 + 1.15i)28-s + (0.0822 + 0.165i)31-s + (0.739 − 0.673i)36-s + (0.136 + 1.47i)37-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)3-s + (−0.850 − 0.526i)4-s + (−0.890 − 0.811i)7-s + (−0.273 + 0.961i)9-s + (0.0922 + 0.995i)12-s + (−1.25 − 1.14i)13-s + (0.445 + 0.895i)16-s + (1.18 − 1.56i)19-s + (−0.111 + 1.20i)21-s + (0.932 + 0.361i)25-s + (0.932 − 0.361i)27-s + (0.329 + 1.15i)28-s + (0.0822 + 0.165i)31-s + (0.739 − 0.673i)36-s + (0.136 + 1.47i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $-0.640 + 0.768i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ -0.640 + 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4235779908\)
\(L(\frac12)\) \(\approx\) \(0.4235779908\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.602 + 0.798i)T \)
103 \( 1 + (-0.0922 - 0.995i)T \)
good2 \( 1 + (0.850 + 0.526i)T^{2} \)
5 \( 1 + (-0.932 - 0.361i)T^{2} \)
7 \( 1 + (0.890 + 0.811i)T + (0.0922 + 0.995i)T^{2} \)
11 \( 1 + (0.850 + 0.526i)T^{2} \)
13 \( 1 + (1.25 + 1.14i)T + (0.0922 + 0.995i)T^{2} \)
17 \( 1 + (-0.739 - 0.673i)T^{2} \)
19 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
23 \( 1 + (0.850 - 0.526i)T^{2} \)
29 \( 1 + (-0.932 - 0.361i)T^{2} \)
31 \( 1 + (-0.0822 - 0.165i)T + (-0.602 + 0.798i)T^{2} \)
37 \( 1 + (-0.136 - 1.47i)T + (-0.982 + 0.183i)T^{2} \)
41 \( 1 + (-0.932 + 0.361i)T^{2} \)
43 \( 1 + (-0.172 + 1.85i)T + (-0.982 - 0.183i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.273 + 0.961i)T^{2} \)
59 \( 1 + (-0.0922 + 0.995i)T^{2} \)
61 \( 1 + (-0.831 - 0.322i)T + (0.739 + 0.673i)T^{2} \)
67 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
71 \( 1 + (-0.932 + 0.361i)T^{2} \)
73 \( 1 + (0.876 - 0.163i)T + (0.932 - 0.361i)T^{2} \)
79 \( 1 + (1.45 + 0.271i)T + (0.932 + 0.361i)T^{2} \)
83 \( 1 + (-0.0922 - 0.995i)T^{2} \)
89 \( 1 + (-0.445 + 0.895i)T^{2} \)
97 \( 1 + (-1.73 + 0.673i)T + (0.739 - 0.673i)T^{2} \)
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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

−11.66101350524184906618653735215, −10.46158802050969909056768221393, −9.951525478504986720454267679252, −8.798565172214713192566790678788, −7.47849579974430418425012093289, −6.81739692030630353188383310319, −5.50693067748970406321471399161, −4.77177887330152903720148791021, −3.00659371621087082806265148110, −0.72641699415523909323945140098, 3.00815477401801007264187115570, 4.16454994610389511621220549632, 5.17018971034988025487703597470, 6.17152686233368839666594682959, 7.47884959226606134943434618025, 8.850365859589208823293548018022, 9.537008982505023249768350653668, 10.04083402063379424294043195882, 11.54472744319000234267811499631, 12.26226051514589456300610938287

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