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Properties

Label 2-338-13.6-c2-0-3
Degree $2$
Conductor $338$
Sign $0.283 + 0.958i$
Analytic cond. $9.20983$
Root an. cond. $3.03477$
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  + (0.366 + 1.36i)2-s + (−2.78 + 4.83i)3-s + (−1.73 + i)4-s + (0.323 − 0.323i)5-s + (−7.62 − 2.04i)6-s + (−2.05 + 7.67i)7-s + (−2 − 1.99i)8-s + (−11.0 − 19.1i)9-s + (0.560 + 0.323i)10-s + (−5.40 + 1.44i)11-s − 11.1i·12-s − 11.2·14-s + (0.661 + 2.46i)15-s + (1.99 − 3.46i)16-s + (1.74 − 1.00i)17-s + (22.1 − 22.1i)18-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.929 + 1.61i)3-s + (−0.433 + 0.250i)4-s + (0.0647 − 0.0647i)5-s + (−1.27 − 0.340i)6-s + (−0.293 + 1.09i)7-s + (−0.250 − 0.249i)8-s + (−1.22 − 2.12i)9-s + (0.0560 + 0.0323i)10-s + (−0.491 + 0.131i)11-s − 0.929i·12-s − 0.803·14-s + (0.0440 + 0.164i)15-s + (0.124 − 0.216i)16-s + (0.102 − 0.0591i)17-s + (1.22 − 1.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.283 + 0.958i$
Analytic conductor: \(9.20983\)
Root analytic conductor: \(3.03477\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1),\ 0.283 + 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.351021 - 0.262226i\)
\(L(\frac12)\) \(\approx\) \(0.351021 - 0.262226i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
13 \( 1 \)
good3 \( 1 + (2.78 - 4.83i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.323 + 0.323i)T - 25iT^{2} \)
7 \( 1 + (2.05 - 7.67i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (5.40 - 1.44i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-1.74 + 1.00i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4.47 + 1.19i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (-15.0 - 8.71i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-8.25 + 14.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (9.27 - 9.27i)T - 961iT^{2} \)
37 \( 1 + (34.0 - 9.12i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (15.7 + 58.6i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-45.2 + 26.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-8.73 - 8.73i)T + 2.20e3iT^{2} \)
53 \( 1 + 23.4T + 2.80e3T^{2} \)
59 \( 1 + (9.13 - 34.0i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (13.9 + 24.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-8.85 - 33.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (73.8 + 19.7i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (18.9 + 18.9i)T + 5.32e3iT^{2} \)
79 \( 1 + 142.T + 6.24e3T^{2} \)
83 \( 1 + (91.7 - 91.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (150. - 40.4i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (108. + 29.1i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.98292662254928711279634755707, −11.04876499633339081672158294222, −10.14766026370848944350146352742, −9.250063776976257054338914227785, −8.678070481764815666785431417818, −7.03812388683350862377567022232, −5.71158184797623231499447304990, −5.43121230716292840278463830349, −4.32149729111494689301721674092, −3.10377561761759161388591726251, 0.22559803151166320965411802436, 1.39163341396058406319092220037, 2.82128773719156698540285726117, 4.55306048342209195308266398038, 5.76352679780396232760669213197, 6.70210475302622153338138275348, 7.50474545411640959320640792507, 8.511666484580022450855430979437, 10.14377226617967422610234682369, 10.82134999581307370237378916995

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