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Properties

Label 2-309-309.14-c0-0-0
Degree $2$
Conductor $309$
Sign $0.983 - 0.181i$
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  + (0.445 + 0.895i)3-s + (−0.273 − 0.961i)4-s + (0.831 − 0.322i)7-s + (−0.602 + 0.798i)9-s + (0.739 − 0.673i)12-s + (−0.510 + 0.197i)13-s + (−0.850 + 0.526i)16-s + (0.0822 − 0.165i)19-s + (0.658 + 0.600i)21-s + (−0.982 + 0.183i)25-s + (−0.982 − 0.183i)27-s + (−0.537 − 0.711i)28-s + (−1.25 + 0.778i)31-s + (0.932 + 0.361i)36-s + (1.37 − 1.25i)37-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)3-s + (−0.273 − 0.961i)4-s + (0.831 − 0.322i)7-s + (−0.602 + 0.798i)9-s + (0.739 − 0.673i)12-s + (−0.510 + 0.197i)13-s + (−0.850 + 0.526i)16-s + (0.0822 − 0.165i)19-s + (0.658 + 0.600i)21-s + (−0.982 + 0.183i)25-s + (−0.982 − 0.183i)27-s + (−0.537 − 0.711i)28-s + (−1.25 + 0.778i)31-s + (0.932 + 0.361i)36-s + (1.37 − 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(309\)    =    \(3 \cdot 103\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(0.154211\)
Root analytic conductor: \(0.392697\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{309} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 309,\ (\ :0),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8582021933\)
\(L(\frac12)\) \(\approx\) \(0.8582021933\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.445 - 0.895i)T \)
103 \( 1 + (-0.739 + 0.673i)T \)
good2 \( 1 + (0.273 + 0.961i)T^{2} \)
5 \( 1 + (0.982 - 0.183i)T^{2} \)
7 \( 1 + (-0.831 + 0.322i)T + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (0.273 + 0.961i)T^{2} \)
13 \( 1 + (0.510 - 0.197i)T + (0.739 - 0.673i)T^{2} \)
17 \( 1 + (-0.932 + 0.361i)T^{2} \)
19 \( 1 + (-0.0822 + 0.165i)T + (-0.602 - 0.798i)T^{2} \)
23 \( 1 + (0.273 - 0.961i)T^{2} \)
29 \( 1 + (0.982 - 0.183i)T^{2} \)
31 \( 1 + (1.25 - 0.778i)T + (0.445 - 0.895i)T^{2} \)
37 \( 1 + (-1.37 + 1.25i)T + (0.0922 - 0.995i)T^{2} \)
41 \( 1 + (0.982 + 0.183i)T^{2} \)
43 \( 1 + (1.45 + 1.32i)T + (0.0922 + 0.995i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.602 + 0.798i)T^{2} \)
59 \( 1 + (-0.739 - 0.673i)T^{2} \)
61 \( 1 + (-1.67 + 0.312i)T + (0.932 - 0.361i)T^{2} \)
67 \( 1 + (1.12 + 0.435i)T + (0.739 + 0.673i)T^{2} \)
71 \( 1 + (0.982 + 0.183i)T^{2} \)
73 \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \)
79 \( 1 + (-0.172 - 1.85i)T + (-0.982 + 0.183i)T^{2} \)
83 \( 1 + (-0.739 + 0.673i)T^{2} \)
89 \( 1 + (0.850 + 0.526i)T^{2} \)
97 \( 1 + (-1.93 - 0.361i)T + (0.932 + 0.361i)T^{2} \)
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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

−11.55230595029454834549589490490, −10.88985095202211612721569337039, −10.02920217596675485077876552646, −9.305746104085438268088656930566, −8.371803745478781027378241718364, −7.22716178830349898011902634157, −5.66148974304000847472976061612, −4.88201569564777905768935026169, −3.88389183131862008755885143852, −2.04574727892630276969122208439, 2.08075804472434536619317061321, 3.34224784128092293674722751179, 4.72713738702971252553956368249, 6.13846450848781255225202170617, 7.45848739336053088044019910801, 7.957024317590855411476059941908, 8.783127502333707737619393283168, 9.791428570435599533969565589528, 11.51854271226176899162336227274, 11.80613781660626067433607435682

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