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Properties

Label 2-338-13.10-c1-0-4
Degree $2$
Conductor $338$
Sign $0.327 + 0.944i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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  + (−0.866 + 0.5i)2-s + (−1.17 − 2.04i)3-s + (0.499 − 0.866i)4-s − 0.890i·5-s + (2.04 + 1.17i)6-s + (3.89 + 2.24i)7-s + 0.999i·8-s + (−1.27 + 2.21i)9-s + (0.445 + 0.770i)10-s + (2.33 − 1.34i)11-s − 2.35·12-s − 4.49·14-s + (−1.81 + 1.04i)15-s + (−0.5 − 0.866i)16-s + (1.79 − 3.10i)17-s − 2.55i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.680 − 1.17i)3-s + (0.249 − 0.433i)4-s − 0.398i·5-s + (0.833 + 0.481i)6-s + (1.47 + 0.849i)7-s + 0.353i·8-s + (−0.425 + 0.737i)9-s + (0.140 + 0.243i)10-s + (0.702 − 0.405i)11-s − 0.680·12-s − 1.20·14-s + (−0.469 + 0.270i)15-s + (−0.125 − 0.216i)16-s + (0.434 − 0.752i)17-s − 0.602i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750285 - 0.533774i\)
\(L(\frac12)\) \(\approx\) \(0.750285 - 0.533774i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (1.17 + 2.04i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.890iT - 5T^{2} \)
7 \( 1 + (-3.89 - 2.24i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.33 + 1.34i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.79 + 3.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.54 + 1.46i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.04 + 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 + (1.30 - 0.753i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.16 + 1.82i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.0854 + 0.148i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.20iT - 47T^{2} \)
53 \( 1 - 6.09T + 53T^{2} \)
59 \( 1 + (2.65 + 1.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.98 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.58 - 5.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.74 + 5.04i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 - 2.81T + 79T^{2} \)
83 \( 1 + 2.93iT - 83T^{2} \)
89 \( 1 + (-10.5 + 6.09i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.2 - 6.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.58153529919560614131770452983, −10.65814228730908752581252334199, −9.133160548627496820187172947913, −8.437378587601681665242390541036, −7.61569427088518024482997097844, −6.57769509497708437101214171383, −5.71007946962903268460824630525, −4.72110328793951695019730483804, −2.18142688680260506465131609312, −0.982213822863960460817008303207, 1.61525478955380364413875893283, 3.76896283144672556192126185218, 4.47970073943377631180853349868, 5.70182033063829178061549681070, 7.12841098743248341476765923329, 8.049880489473158238896101994372, 9.167011730117438981417541191521, 10.26421302857007877748862151288, 10.61932959092735894442898738635, 11.39999352790739938622477006310

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