Properties

Label 2-309-309.23-c0-0-0
Degree $2$
Conductor $309$
Sign $0.998 - 0.0524i$
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  + (−0.850 − 0.526i)3-s + (−0.602 + 0.798i)4-s + (1.67 − 0.312i)7-s + (0.445 + 0.895i)9-s + (0.932 − 0.361i)12-s + (1.18 − 0.221i)13-s + (−0.273 − 0.961i)16-s + (−1.25 + 0.778i)19-s + (−1.58 − 0.614i)21-s + (0.0922 + 0.995i)25-s + (0.0922 − 0.995i)27-s + (−0.757 + 1.52i)28-s + (−0.510 − 1.79i)31-s + (−0.982 − 0.183i)36-s + (−1.83 + 0.710i)37-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)3-s + (−0.602 + 0.798i)4-s + (1.67 − 0.312i)7-s + (0.445 + 0.895i)9-s + (0.932 − 0.361i)12-s + (1.18 − 0.221i)13-s + (−0.273 − 0.961i)16-s + (−1.25 + 0.778i)19-s + (−1.58 − 0.614i)21-s + (0.0922 + 0.995i)25-s + (0.0922 − 0.995i)27-s + (−0.757 + 1.52i)28-s + (−0.510 − 1.79i)31-s + (−0.982 − 0.183i)36-s + (−1.83 + 0.710i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.998 - 0.0524i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.998 - 0.0524i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6381602227\)
\(L(\frac12)\) \(\approx\) \(0.6381602227\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.850 + 0.526i)T \)
103 \( 1 + (-0.932 + 0.361i)T \)
good2 \( 1 + (0.602 - 0.798i)T^{2} \)
5 \( 1 + (-0.0922 - 0.995i)T^{2} \)
7 \( 1 + (-1.67 + 0.312i)T + (0.932 - 0.361i)T^{2} \)
11 \( 1 + (0.602 - 0.798i)T^{2} \)
13 \( 1 + (-1.18 + 0.221i)T + (0.932 - 0.361i)T^{2} \)
17 \( 1 + (0.982 - 0.183i)T^{2} \)
19 \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \)
23 \( 1 + (0.602 + 0.798i)T^{2} \)
29 \( 1 + (-0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.510 + 1.79i)T + (-0.850 + 0.526i)T^{2} \)
37 \( 1 + (1.83 - 0.710i)T + (0.739 - 0.673i)T^{2} \)
41 \( 1 + (-0.0922 + 0.995i)T^{2} \)
43 \( 1 + (-0.172 - 0.0666i)T + (0.739 + 0.673i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.445 + 0.895i)T^{2} \)
59 \( 1 + (-0.932 - 0.361i)T^{2} \)
61 \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \)
67 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
71 \( 1 + (-0.0922 + 0.995i)T^{2} \)
73 \( 1 + (0.404 - 0.368i)T + (0.0922 - 0.995i)T^{2} \)
79 \( 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2} \)
83 \( 1 + (-0.932 + 0.361i)T^{2} \)
89 \( 1 + (0.273 - 0.961i)T^{2} \)
97 \( 1 + (-0.0170 + 0.183i)T + (-0.982 - 0.183i)T^{2} \)
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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

−11.80894465724069239606319471569, −11.20200324252576302809585622427, −10.39031535540935810838677098001, −8.752456090034662549525632367606, −8.061491590482139052073949704842, −7.31601429701644694993377435300, −5.91640941705788943828832392585, −4.85438055086274944963101221197, −3.90286763445706110000788804264, −1.69601427373282668613549886870, 1.55615360331185788181196393335, 4.12697033473770647727916457883, 4.90724342423505238733144065902, 5.71774428695125906409865060186, 6.79453629879544450608147051306, 8.563181429018852799138341846363, 8.913172586861299496329061931624, 10.49737177754840357752338282034, 10.76110619507991361836729649595, 11.65326649728986215776347078424

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