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Properties

Label 2-308-11.3-c1-0-3
Degree $2$
Conductor $308$
Sign $0.141 - 0.989i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.813 + 2.50i)3-s + (3.11 + 2.26i)5-s + (0.309 − 0.951i)7-s + (−3.18 + 2.31i)9-s + (0.390 − 3.29i)11-s + (1.46 − 1.06i)13-s + (−3.13 + 9.64i)15-s + (−3.96 − 2.88i)17-s + (−1.95 − 6.02i)19-s + 2.63·21-s − 6.51·23-s + (3.03 + 9.34i)25-s + (−1.99 − 1.44i)27-s + (−1.67 + 5.15i)29-s + (2.18 − 1.58i)31-s + ⋯
L(s)  = 1  + (0.469 + 1.44i)3-s + (1.39 + 1.01i)5-s + (0.116 − 0.359i)7-s + (−1.06 + 0.770i)9-s + (0.117 − 0.993i)11-s + (0.405 − 0.294i)13-s + (−0.809 + 2.49i)15-s + (−0.962 − 0.699i)17-s + (−0.448 − 1.38i)19-s + 0.574·21-s − 1.35·23-s + (0.607 + 1.86i)25-s + (−0.382 − 0.278i)27-s + (−0.311 + 0.957i)29-s + (0.391 − 0.284i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.141 - 0.989i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36265 + 1.18180i\)
\(L(\frac12)\) \(\approx\) \(1.36265 + 1.18180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.390 + 3.29i)T \)
good3 \( 1 + (-0.813 - 2.50i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-3.11 - 2.26i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.46 + 1.06i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.96 + 2.88i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.95 + 6.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 + (1.67 - 5.15i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.18 + 1.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.639 + 1.96i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.12 - 9.62i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.13T + 43T^{2} \)
47 \( 1 + (3.38 + 10.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.76 + 4.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.01 - 9.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.71 + 4.87i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.67T + 67T^{2} \)
71 \( 1 + (-7.51 - 5.46i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.662 + 2.03i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.98 - 5.80i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.41 - 6.11i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 + (-3.70 + 2.69i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.32001902553858359681320195735, −10.79931574067493091816004930129, −10.05440302537639258048356616101, −9.307369133938336216512948020844, −8.502253728529872611739544432065, −6.86753261490065516362822543359, −5.92230765499905604032401714176, −4.76883345800501209612237374449, −3.47171590565886224653320859654, −2.47370381671474146138604215665, 1.68931270630486257333841305903, 2.09935591298823629906236776184, 4.38235763230994302386145492877, 5.91678745348335461198561397818, 6.38535738629890405353186073960, 7.79031633674584427753204741808, 8.565941491982133493907135851291, 9.400333541809513232714537944818, 10.37255418444263121161105020930, 12.04867332112202242458029055634

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