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Properties

Label 2-309-309.56-c0-0-1
Degree $2$
Conductor $309$
Sign $0.957 - 0.287i$
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  + (0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s − i·8-s + 9-s + 0.999·10-s + (−0.866 + 0.5i)11-s + 0.999i·14-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)6-s + (0.5 + 0.866i)7-s − i·8-s + 9-s + 0.999·10-s + (−0.866 + 0.5i)11-s + 0.999i·14-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9600255824\)
\(L(\frac12)\) \(\approx\) \(0.9600255824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
103 \( 1 + T \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + 2iT - T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−12.45534161045874148633947490910, −11.06092065881778440881071930999, −10.18205212697811268598128540579, −9.312080654328043383870166357982, −8.005020077092955937159925138564, −6.50161125793633505369815806766, −5.91955602453838400500700878125, −5.06384152626733809355963094977, −4.39624143448930314733982286000, −1.99031898283215354411522817944, 2.12258877966992706501097139339, 3.68952409233427246372690665657, 4.91233275782243722896651321198, 5.57716064173532486415388486645, 6.81183079253806388248110995934, 7.81738152233174997927230057950, 9.343797215648269342673598982960, 10.54736478860807642989992341109, 11.05735634184755926265926199902, 11.70059751079575440743216931734

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