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Properties

Label 2-378-63.41-c1-0-5
Degree $2$
Conductor $378$
Sign $-0.0232 + 0.999i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
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 + (0.499 − 0.866i)4-s + (0.895 − 1.55i)5-s + (−2.30 − 1.30i)7-s + 0.999i·8-s + 1.79i·10-s + (2.07 − 1.20i)11-s + (−4.23 − 2.44i)13-s + (2.64 − 0.0213i)14-s + (−0.5 − 0.866i)16-s − 3.66·17-s − 3.01i·19-s + (−0.895 − 1.55i)20-s + (−1.20 + 2.07i)22-s + (−3.26 − 1.88i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.400 − 0.693i)5-s + (−0.870 − 0.492i)7-s + 0.353i·8-s + 0.566i·10-s + (0.627 − 0.362i)11-s + (−1.17 − 0.678i)13-s + (0.707 − 0.00571i)14-s + (−0.125 − 0.216i)16-s − 0.888·17-s − 0.692i·19-s + (−0.200 − 0.346i)20-s + (−0.256 + 0.443i)22-s + (−0.680 − 0.392i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0232 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0232 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.0232 + 0.999i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.0232 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521228 - 0.533504i\)
\(L(\frac12)\) \(\approx\) \(0.521228 - 0.533504i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (2.30 + 1.30i)T \)
good5 \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.07 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.23 + 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 + 3.01iT - 19T^{2} \)
23 \( 1 + (3.26 + 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.68 + 3.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.02 + 2.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (-4.04 + 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.48 + 6.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.56 + 4.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (7.29 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.81 + 5.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.285 - 0.493i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.00 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + (-4.77 + 2.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.91180465170534941348828919314, −9.949675049615695260822933469081, −9.341283035346819633257512957020, −8.494112381244965487463221539248, −7.33038324651215629582442667976, −6.49154718538628936504869253242, −5.44914401610847586097375797952, −4.21164709465337088199989575395, −2.50878354181396843450313673911, −0.58552571975631065600301525442, 2.04020815881690928892311926232, 3.08304445779108362641185327857, 4.53333520756769638949979436372, 6.25267925097910091894376931973, 6.75898322832764142968480086335, 7.921936176738876573712260350599, 9.238801068101947419256482182997, 9.653642994320131493781631531844, 10.50557869829454987388102025017, 11.57942374073258466201819439371

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