↑

Properties

Label 2-309-309.236-c0-0-0
Degree $2$
Conductor $309$
Sign $0.846 - 0.532i$
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  + (0.932 + 0.361i)3-s + (0.0922 + 0.995i)4-s + (−1.12 − 1.48i)7-s + (0.739 + 0.673i)9-s + (−0.273 + 0.961i)12-s + (−0.111 − 0.147i)13-s + (−0.982 + 0.183i)16-s + (−1.58 + 0.614i)19-s + (−0.510 − 1.79i)21-s + (0.445 − 0.895i)25-s + (0.445 + 0.895i)27-s + (1.37 − 1.25i)28-s + (0.538 − 0.100i)31-s + (−0.602 + 0.798i)36-s + (0.329 − 1.15i)37-s + ⋯
L(s)  = 1  + (0.932 + 0.361i)3-s + (0.0922 + 0.995i)4-s + (−1.12 − 1.48i)7-s + (0.739 + 0.673i)9-s + (−0.273 + 0.961i)12-s + (−0.111 − 0.147i)13-s + (−0.982 + 0.183i)16-s + (−1.58 + 0.614i)19-s + (−0.510 − 1.79i)21-s + (0.445 − 0.895i)25-s + (0.445 + 0.895i)27-s + (1.37 − 1.25i)28-s + (0.538 − 0.100i)31-s + (−0.602 + 0.798i)36-s + (0.329 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9396841670\)
\(L(\frac12)\) \(\approx\) \(0.9396841670\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.932 - 0.361i)T \)
103 \( 1 + (0.273 - 0.961i)T \)
good2 \( 1 + (-0.0922 - 0.995i)T^{2} \)
5 \( 1 + (-0.445 + 0.895i)T^{2} \)
7 \( 1 + (1.12 + 1.48i)T + (-0.273 + 0.961i)T^{2} \)
11 \( 1 + (-0.0922 - 0.995i)T^{2} \)
13 \( 1 + (0.111 + 0.147i)T + (-0.273 + 0.961i)T^{2} \)
17 \( 1 + (0.602 + 0.798i)T^{2} \)
19 \( 1 + (1.58 - 0.614i)T + (0.739 - 0.673i)T^{2} \)
23 \( 1 + (-0.0922 + 0.995i)T^{2} \)
29 \( 1 + (-0.445 + 0.895i)T^{2} \)
31 \( 1 + (-0.538 + 0.100i)T + (0.932 - 0.361i)T^{2} \)
37 \( 1 + (-0.329 + 1.15i)T + (-0.850 - 0.526i)T^{2} \)
41 \( 1 + (-0.445 - 0.895i)T^{2} \)
43 \( 1 + (0.243 + 0.857i)T + (-0.850 + 0.526i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.739 + 0.673i)T^{2} \)
59 \( 1 + (0.273 + 0.961i)T^{2} \)
61 \( 1 + (0.876 - 1.75i)T + (-0.602 - 0.798i)T^{2} \)
67 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
71 \( 1 + (-0.445 - 0.895i)T^{2} \)
73 \( 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2} \)
79 \( 1 + (-1.02 + 0.634i)T + (0.445 - 0.895i)T^{2} \)
83 \( 1 + (0.273 - 0.961i)T^{2} \)
89 \( 1 + (0.982 + 0.183i)T^{2} \)
97 \( 1 + (-0.397 - 0.798i)T + (-0.602 + 0.798i)T^{2} \)
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

−12.31569060497030174743071193700, −10.72676519551658863596847162533, −10.20874829025314056817697916765, −9.108977568543425082256395149114, −8.181443832742080989463405906122, −7.33192034374296954730545634974, −6.49993596061875050029326002334, −4.30736528328072380122476622920, −3.77206750305286369927500352203, −2.59858853439298965045387583941, 2.05444099293174675285597566670, 3.08336689044397880433616761736, 4.82767147543340809819951420477, 6.22913292760306596889452002153, 6.67743208978662796260658798064, 8.275431249288019801535924985805, 9.207136828037597501933667044614, 9.602167831389700319157184571826, 10.78938846037624845880768967828, 12.01955567923549501584396834956

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