↑

Properties

Label 2-309-309.179-c0-0-0
Degree $2$
Conductor $309$
Sign $0.482 - 0.875i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.739 + 0.673i)3-s + (−0.982 + 0.183i)4-s + (−0.404 + 1.42i)7-s + (0.0922 + 0.995i)9-s + (−0.850 − 0.526i)12-s + (0.538 − 1.89i)13-s + (0.932 − 0.361i)16-s + (0.658 − 0.600i)19-s + (−1.25 + 0.778i)21-s + (−0.602 − 0.798i)25-s + (−0.602 + 0.798i)27-s + (0.136 − 1.47i)28-s + (−1.58 + 0.614i)31-s + (−0.273 − 0.961i)36-s + (0.465 + 0.288i)37-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)3-s + (−0.982 + 0.183i)4-s + (−0.404 + 1.42i)7-s + (0.0922 + 0.995i)9-s + (−0.850 − 0.526i)12-s + (0.538 − 1.89i)13-s + (0.932 − 0.361i)16-s + (0.658 − 0.600i)19-s + (−1.25 + 0.778i)21-s + (−0.602 − 0.798i)25-s + (−0.602 + 0.798i)27-s + (0.136 − 1.47i)28-s + (−1.58 + 0.614i)31-s + (−0.273 − 0.961i)36-s + (0.465 + 0.288i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.482 - 0.875i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.482 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7875806856\)
\(L(\frac12)\) \(\approx\) \(0.7875806856\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.739 - 0.673i)T \)
103 \( 1 + (0.850 + 0.526i)T \)
good2 \( 1 + (0.982 - 0.183i)T^{2} \)
5 \( 1 + (0.602 + 0.798i)T^{2} \)
7 \( 1 + (0.404 - 1.42i)T + (-0.850 - 0.526i)T^{2} \)
11 \( 1 + (0.982 - 0.183i)T^{2} \)
13 \( 1 + (-0.538 + 1.89i)T + (-0.850 - 0.526i)T^{2} \)
17 \( 1 + (0.273 - 0.961i)T^{2} \)
19 \( 1 + (-0.658 + 0.600i)T + (0.0922 - 0.995i)T^{2} \)
23 \( 1 + (0.982 + 0.183i)T^{2} \)
29 \( 1 + (0.602 + 0.798i)T^{2} \)
31 \( 1 + (1.58 - 0.614i)T + (0.739 - 0.673i)T^{2} \)
37 \( 1 + (-0.465 - 0.288i)T + (0.445 + 0.895i)T^{2} \)
41 \( 1 + (0.602 - 0.798i)T^{2} \)
43 \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.0922 + 0.995i)T^{2} \)
59 \( 1 + (0.850 - 0.526i)T^{2} \)
61 \( 1 + (1.12 + 1.48i)T + (-0.273 + 0.961i)T^{2} \)
67 \( 1 + (0.0505 + 0.177i)T + (-0.850 + 0.526i)T^{2} \)
71 \( 1 + (0.602 - 0.798i)T^{2} \)
73 \( 1 + (-0.831 - 1.66i)T + (-0.602 + 0.798i)T^{2} \)
79 \( 1 + (0.243 - 0.489i)T + (-0.602 - 0.798i)T^{2} \)
83 \( 1 + (0.850 + 0.526i)T^{2} \)
89 \( 1 + (-0.932 - 0.361i)T^{2} \)
97 \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.38218220347235528074810917281, −10.95140664161685825032121579389, −9.933188261336636290045122135672, −9.195301172129841668029023278072, −8.505647762336045904972405390635, −7.73500131443855944892512441097, −5.75845559360256906075967844815, −5.10750870487869113079581839819, −3.64790485077002313220877868367, −2.74873780200376509986631074823, 1.47116311445502777353997372343, 3.64252366939567837419407403303, 4.22758766611847168518148531894, 5.99508416995555671361693453387, 7.13148316670685032788513440796, 7.84649066118196175217544528737, 9.206819302172262542043182912956, 9.464484728230653354779439448570, 10.74523262210507026964211158023, 11.90205293946413255757338783735

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