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Properties

Label 2-308-11.2-c2-0-2
Degree $2$
Conductor $308$
Sign $-0.653 - 0.757i$
Analytic cond. $8.39239$
Root an. cond. $2.89696$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.251 − 0.182i)3-s + (2.69 + 8.30i)5-s + (−1.55 + 2.14i)7-s + (−2.75 + 8.46i)9-s + (−7.36 − 8.17i)11-s + (−4.10 − 1.33i)13-s + (2.19 + 1.59i)15-s + (−21.4 + 6.96i)17-s + (−12.2 − 16.8i)19-s + 0.821i·21-s + 29.3·23-s + (−41.3 + 30.0i)25-s + (1.71 + 5.28i)27-s + (8.91 − 12.2i)29-s + (−9.46 + 29.1i)31-s + ⋯
L(s)  = 1  + (0.0837 − 0.0608i)3-s + (0.539 + 1.66i)5-s + (−0.222 + 0.305i)7-s + (−0.305 + 0.940i)9-s + (−0.669 − 0.742i)11-s + (−0.315 − 0.102i)13-s + (0.146 + 0.106i)15-s + (−1.26 + 0.409i)17-s + (−0.646 − 0.889i)19-s + 0.0391i·21-s + 1.27·23-s + (−1.65 + 1.20i)25-s + (0.0636 + 0.195i)27-s + (0.307 − 0.423i)29-s + (−0.305 + 0.939i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.653 - 0.757i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ -0.653 - 0.757i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.512093 + 1.11783i\)
\(L(\frac12)\) \(\approx\) \(0.512093 + 1.11783i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.55 - 2.14i)T \)
11 \( 1 + (7.36 + 8.17i)T \)
good3 \( 1 + (-0.251 + 0.182i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (-2.69 - 8.30i)T + (-20.2 + 14.6i)T^{2} \)
13 \( 1 + (4.10 + 1.33i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (21.4 - 6.96i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (12.2 + 16.8i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 29.3T + 529T^{2} \)
29 \( 1 + (-8.91 + 12.2i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (9.46 - 29.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-49.0 - 35.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-21.9 - 30.2i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 42.9iT - 1.84e3T^{2} \)
47 \( 1 + (19.3 - 14.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-14.9 + 45.8i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-79.3 - 57.6i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (77.7 - 25.2i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 45.5T + 4.48e3T^{2} \)
71 \( 1 + (11.9 + 36.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-22.8 + 31.4i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-16.7 - 5.44i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (49.3 - 16.0i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 117.T + 7.92e3T^{2} \)
97 \( 1 + (10.3 - 31.8i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.33590398576154663981215199295, −10.94506827770928169063166641473, −10.20739522925263886982855250274, −9.001509111010649268188514512523, −7.932558257004659749754440320680, −6.82940010626564202516649744640, −6.11765712454619932360991029572, −4.85173899258058080734518101772, −2.96880673144272039702316996040, −2.41926096863239235410930336951, 0.55336237077616444840444149714, 2.20679122021699118831803905832, 4.08830263252325713917335319952, 4.99195848225596273972034410319, 6.05179851406614722312507463149, 7.29016912664017140508156642683, 8.575809101491144215214906732083, 9.212074724191653760767230531545, 9.928144018605015152972212385738, 11.17861097095092261406057709338

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