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Properties

Label 2-308-11.8-c2-0-4
Degree $2$
Conductor $308$
Sign $0.966 + 0.257i$
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  + (−1.08 − 3.34i)3-s + (4.48 + 3.25i)5-s + (2.51 + 0.817i)7-s + (−2.72 + 1.97i)9-s + (−0.179 + 10.9i)11-s + (14.1 + 19.4i)13-s + (6.02 − 18.5i)15-s + (11.0 − 15.2i)17-s + (−6.69 + 2.17i)19-s − 9.30i·21-s − 10.7·23-s + (1.77 + 5.46i)25-s + (−16.0 − 11.6i)27-s + (34.1 + 11.1i)29-s + (32.7 − 23.7i)31-s + ⋯
L(s)  = 1  + (−0.362 − 1.11i)3-s + (0.897 + 0.651i)5-s + (0.359 + 0.116i)7-s + (−0.302 + 0.219i)9-s + (−0.0163 + 0.999i)11-s + (1.08 + 1.49i)13-s + (0.401 − 1.23i)15-s + (0.651 − 0.896i)17-s + (−0.352 + 0.114i)19-s − 0.443i·21-s − 0.468·23-s + (0.0710 + 0.218i)25-s + (−0.593 − 0.431i)27-s + (1.17 + 0.382i)29-s + (1.05 − 0.767i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.966 + 0.257i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ 0.966 + 0.257i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.83560 - 0.240851i\)
\(L(\frac12)\) \(\approx\) \(1.83560 - 0.240851i\)
\(L(2)\) 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 \( 1 + (-2.51 - 0.817i)T \)
11 \( 1 + (0.179 - 10.9i)T \)
good3 \( 1 + (1.08 + 3.34i)T + (-7.28 + 5.29i)T^{2} \)
5 \( 1 + (-4.48 - 3.25i)T + (7.72 + 23.7i)T^{2} \)
13 \( 1 + (-14.1 - 19.4i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-11.0 + 15.2i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (6.69 - 2.17i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 10.7T + 529T^{2} \)
29 \( 1 + (-34.1 - 11.1i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-32.7 + 23.7i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (7.91 - 24.3i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-1.93 + 0.628i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 72.8iT - 1.84e3T^{2} \)
47 \( 1 + (-2.50 - 7.70i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-69.7 + 50.6i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (22.6 - 69.6i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (38.6 - 53.1i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 + 7.80T + 4.48e3T^{2} \)
71 \( 1 + (12.2 + 8.92i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (110. + 35.7i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-18.0 - 24.8i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (3.64 - 5.01i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 139.T + 7.92e3T^{2} \)
97 \( 1 + (4.34 - 3.15i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.84426529038265258686664155862, −10.49262767496693872577652856620, −9.680164990849413262252665317847, −8.516308564185046144330658856595, −7.23059990842256575000120636856, −6.63840479071679568391695756570, −5.81678757957278315167201741576, −4.32872382162493120342620772339, −2.39792945389569374668330914596, −1.42070127461666213546088658311, 1.16903113425968207409793671772, 3.24994068371153504889591286884, 4.48189511805147716235800232218, 5.57724984513219979888226137487, 6.06347188887035566507094080906, 8.071578131767944822631601685959, 8.695938855034969393979810839275, 9.904023677937215480443456009598, 10.48390732084227498391355600605, 11.16683381880317477733202612526

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