Properties

Label 2-378-7.2-c3-0-30
Degree $2$
Conductor $378$
Sign $-0.999 - 0.0290i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (8.00 − 13.8i)5-s + (1.70 − 18.4i)7-s − 7.99·8-s + (−16.0 − 27.7i)10-s + (−7.89 − 13.6i)11-s − 50.4·13-s + (−30.2 − 21.4i)14-s + (−8 + 13.8i)16-s + (18.4 + 32.0i)17-s + (44.6 − 77.4i)19-s − 64.0·20-s − 31.5·22-s + (−30.6 + 53.1i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.716 − 1.24i)5-s + (0.0922 − 0.995i)7-s − 0.353·8-s + (−0.506 − 0.877i)10-s + (−0.216 − 0.374i)11-s − 1.07·13-s + (−0.577 − 0.408i)14-s + (−0.125 + 0.216i)16-s + (0.263 + 0.456i)17-s + (0.539 − 0.934i)19-s − 0.716·20-s − 0.306·22-s + (−0.278 + 0.481i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.0290i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.999 - 0.0290i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.948996648\)
\(L(\frac12)\) \(\approx\) \(1.948996648\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 + (-1.70 + 18.4i)T \)
good5 \( 1 + (-8.00 + 13.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.89 + 13.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 50.4T + 2.19e3T^{2} \)
17 \( 1 + (-18.4 - 32.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-44.6 + 77.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (30.6 - 53.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 126.T + 2.43e4T^{2} \)
31 \( 1 + (-129. - 225. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-123. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 516.T + 6.89e4T^{2} \)
43 \( 1 + 34.2T + 7.95e4T^{2} \)
47 \( 1 + (182. - 316. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-23.5 - 40.8i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (358. + 620. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-348. + 603. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (55.3 + 95.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 607.T + 3.57e5T^{2} \)
73 \( 1 + (-140. - 242. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-441. + 764. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 63.1T + 5.71e5T^{2} \)
89 \( 1 + (-337. + 585. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 976.T + 9.12e5T^{2} \)
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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

−10.36248731848865956667640762284, −9.757421151622895103417726986117, −8.837961907681552891569481987701, −7.77036009016216060621119123716, −6.48443922562348461158271404526, −5.13891382667963143889537145158, −4.69460247562870259117520781086, −3.23407876064048439419595253761, −1.68911399444701172437930817569, −0.57686313330881880613319369787, 2.24534000652016275993402283988, 3.08040547777373522846601859957, 4.77087438290865956155777571270, 5.75225967812683539657255920725, 6.55231421351646382584671014642, 7.46980426356711314634414865434, 8.460837590551585558255544457213, 9.838013815623014901374937924035, 10.11867589486905334608677881882, 11.68677128289464211881384807649

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