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Properties

Label 2-308-11.7-c2-0-7
Degree $2$
Conductor $308$
Sign $0.602 + 0.798i$
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  + (0.450 − 1.38i)3-s + (6.79 − 4.93i)5-s + (2.51 − 0.817i)7-s + (5.56 + 4.04i)9-s + (10.1 − 4.29i)11-s + (−13.0 + 17.9i)13-s + (−3.78 − 11.6i)15-s + (−17.2 − 23.7i)17-s + (25.3 + 8.24i)19-s − 3.85i·21-s − 27.9·23-s + (14.0 − 43.3i)25-s + (18.7 − 13.5i)27-s + (−13.8 + 4.50i)29-s + (36.7 + 26.6i)31-s + ⋯
L(s)  = 1  + (0.150 − 0.461i)3-s + (1.35 − 0.987i)5-s + (0.359 − 0.116i)7-s + (0.618 + 0.449i)9-s + (0.920 − 0.390i)11-s + (−1.00 + 1.38i)13-s + (−0.252 − 0.775i)15-s + (−1.01 − 1.39i)17-s + (1.33 + 0.433i)19-s − 0.183i·21-s − 1.21·23-s + (0.563 − 1.73i)25-s + (0.692 − 0.503i)27-s + (−0.477 + 0.155i)29-s + (1.18 + 0.860i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.602 + 0.798i$
Analytic conductor: \(8.39239\)
Root analytic conductor: \(2.89696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1),\ 0.602 + 0.798i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.06657 - 1.02978i\)
\(L(\frac12)\) \(\approx\) \(2.06657 - 1.02978i\)
\(L(2)\) 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 \( 1 + (-10.1 + 4.29i)T \)
good3 \( 1 + (-0.450 + 1.38i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-6.79 + 4.93i)T + (7.72 - 23.7i)T^{2} \)
13 \( 1 + (13.0 - 17.9i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (17.2 + 23.7i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-25.3 - 8.24i)T + (292. + 212. i)T^{2} \)
23 \( 1 + 27.9T + 529T^{2} \)
29 \( 1 + (13.8 - 4.50i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-36.7 - 26.6i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (7.03 + 21.6i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (24.9 + 8.09i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 16.5iT - 1.84e3T^{2} \)
47 \( 1 + (-10.2 + 31.6i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (35.6 + 25.8i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-2.95 - 9.09i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-31.7 - 43.6i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 82.8T + 4.48e3T^{2} \)
71 \( 1 + (-17.5 + 12.7i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (113. - 36.9i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (29.5 - 40.7i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-16.1 - 22.2i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 5.78T + 7.92e3T^{2} \)
97 \( 1 + (-99.5 - 72.3i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.67358005371738284257654384636, −10.08638028417256142027629521861, −9.442866864119943458532108380071, −8.710123408296755683174694441968, −7.36295949273012052665575873087, −6.51317489085083288242910710134, −5.21653735446786598114480245248, −4.41957689577626650882668667112, −2.23114098213762158237898558246, −1.32745424839639365111650313577, 1.77333224051375229558863900184, 3.06326881323821549138693032565, 4.45916665532532271832317030334, 5.78804858911499451486951800607, 6.60382115067384399182157019053, 7.68515462126571011748430987243, 9.101264448478699493677937318966, 10.04155343894423277344962851933, 10.21600871332243483919947546101, 11.50467916018296949570532944490

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