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Properties

Label 2-308-11.8-c2-0-1
Degree $2$
Conductor $308$
Sign $-0.728 + 0.685i$
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  + (1.68 + 5.19i)3-s + (−6.32 − 4.59i)5-s + (2.51 + 0.817i)7-s + (−16.8 + 12.2i)9-s + (−9.25 − 5.94i)11-s + (−7.64 − 10.5i)13-s + (13.2 − 40.6i)15-s + (−12.4 + 17.0i)17-s + (−2.86 + 0.931i)19-s + 14.4i·21-s − 26.2·23-s + (11.1 + 34.4i)25-s + (−52.5 − 38.1i)27-s + (29.0 + 9.45i)29-s + (22.7 − 16.5i)31-s + ⋯
L(s)  = 1  + (0.562 + 1.73i)3-s + (−1.26 − 0.919i)5-s + (0.359 + 0.116i)7-s + (−1.87 + 1.36i)9-s + (−0.841 − 0.540i)11-s + (−0.588 − 0.809i)13-s + (0.880 − 2.71i)15-s + (−0.730 + 1.00i)17-s + (−0.150 + 0.0490i)19-s + 0.688i·21-s − 1.14·23-s + (0.447 + 1.37i)25-s + (−1.94 − 1.41i)27-s + (1.00 + 0.325i)29-s + (0.733 − 0.532i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.728 + 0.685i$
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: \((2,\ 308,\ (\ :1),\ -0.728 + 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0807496 - 0.203686i\)
\(L(\frac12)\) \(\approx\) \(0.0807496 - 0.203686i\)
\(L(2)\) 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 \( 1 + (9.25 + 5.94i)T \)
good3 \( 1 + (-1.68 - 5.19i)T + (-7.28 + 5.29i)T^{2} \)
5 \( 1 + (6.32 + 4.59i)T + (7.72 + 23.7i)T^{2} \)
13 \( 1 + (7.64 + 10.5i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (12.4 - 17.0i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (2.86 - 0.931i)T + (292. - 212. i)T^{2} \)
23 \( 1 + 26.2T + 529T^{2} \)
29 \( 1 + (-29.0 - 9.45i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-22.7 + 16.5i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-14.3 + 44.1i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (47.1 - 15.3i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 2.55iT - 1.84e3T^{2} \)
47 \( 1 + (-11.8 - 36.3i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (62.9 - 45.7i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (5.26 - 16.2i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (56.6 - 77.9i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 79.8T + 4.48e3T^{2} \)
71 \( 1 + (18.5 + 13.4i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (0.404 + 0.131i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (60.3 + 83.0i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (95.3 - 131. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 36.9T + 7.92e3T^{2} \)
97 \( 1 + (60.6 - 44.0i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.88909624975932532441074089012, −10.88183866475958821039063253685, −10.28646308239917593365298016409, −9.145795725801189868481106649430, −8.174504902482744503954518216645, −8.035497779504265652892327028259, −5.67501228805593263562582614125, −4.65381967145249810702750661363, −4.07493320633895554111259441275, −2.85022662127640399407982104531, 0.092065022540923803153714762518, 2.09791103955531594396200343193, 3.06152465031769555174110645717, 4.64183828530415863810639347546, 6.58785541598113825366423371882, 7.06134489751800050214101095584, 7.87995981001609038309229992443, 8.454063621456343840128610789414, 9.995538997651092268540667911825, 11.37904451551980652847195556698

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