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Properties

Label 2-308-11.8-c2-0-5
Degree $2$
Conductor $308$
Sign $0.986 - 0.161i$
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  + (0.476 + 1.46i)3-s + (−4.13 − 3.00i)5-s + (2.51 + 0.817i)7-s + (5.35 − 3.88i)9-s + (4.35 + 10.1i)11-s + (0.424 + 0.584i)13-s + (2.43 − 7.50i)15-s + (5.04 − 6.94i)17-s + (19.6 − 6.38i)19-s + 4.08i·21-s + 41.2·23-s + (0.351 + 1.08i)25-s + (19.5 + 14.1i)27-s + (24.2 + 7.88i)29-s + (−35.2 + 25.6i)31-s + ⋯
L(s)  = 1  + (0.158 + 0.489i)3-s + (−0.827 − 0.601i)5-s + (0.359 + 0.116i)7-s + (0.594 − 0.432i)9-s + (0.395 + 0.918i)11-s + (0.0326 + 0.0449i)13-s + (0.162 − 0.500i)15-s + (0.296 − 0.408i)17-s + (1.03 − 0.336i)19-s + 0.194i·21-s + 1.79·23-s + (0.0140 + 0.0432i)25-s + (0.722 + 0.524i)27-s + (0.836 + 0.271i)29-s + (−1.13 + 0.826i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.986 - 0.161i$
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: \((2,\ 308,\ (\ :1),\ 0.986 - 0.161i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70904 + 0.139096i\)
\(L(\frac12)\) \(\approx\) \(1.70904 + 0.139096i\)
\(L(2)\) 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 \( 1 + (-4.35 - 10.1i)T \)
good3 \( 1 + (-0.476 - 1.46i)T + (-7.28 + 5.29i)T^{2} \)
5 \( 1 + (4.13 + 3.00i)T + (7.72 + 23.7i)T^{2} \)
13 \( 1 + (-0.424 - 0.584i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (-5.04 + 6.94i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-19.6 + 6.38i)T + (292. - 212. i)T^{2} \)
23 \( 1 - 41.2T + 529T^{2} \)
29 \( 1 + (-24.2 - 7.88i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (35.2 - 25.6i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (0.262 - 0.809i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-42.5 + 13.8i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 48.7iT - 1.84e3T^{2} \)
47 \( 1 + (16.2 + 50.1i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (21.7 - 15.8i)T + (868. - 2.67e3i)T^{2} \)
59 \( 1 + (-11.5 + 35.4i)T + (-2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (30.8 - 42.4i)T + (-1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 94.7T + 4.48e3T^{2} \)
71 \( 1 + (53.9 + 39.2i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (90.8 + 29.5i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (68.0 + 93.7i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-73.6 + 101. i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 44.6T + 7.92e3T^{2} \)
97 \( 1 + (145. - 105. i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.63148722408974783368021262313, −10.55345236217329032715592952199, −9.433901455903203358385477819542, −8.889508465759773530302683178369, −7.61560585438469088343253570313, −6.87341922309437376309191727191, −5.08813696197630811198256395169, −4.43735959435248248410742855267, −3.23384630380340371623823151274, −1.16119678477496888388407771839, 1.17294831119309335197068486577, 2.97451749568035204824316070463, 4.08842672980818553463069297980, 5.50087167635877400666693676319, 6.86184246638344293984992411484, 7.56933495978024290521758630531, 8.347436229125655960134192147642, 9.563947679028443736542984064778, 10.85746263774568444188636557485, 11.28869529677699169197391174693

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