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Properties

Label 2-378-63.38-c3-0-12
Degree $2$
Conductor $378$
Sign $0.327 - 0.944i$
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  + 2i·2-s − 4·4-s + (3.13 − 5.42i)5-s + (13.5 + 12.6i)7-s − 8i·8-s + (10.8 + 6.26i)10-s + (−30.9 + 17.8i)11-s + (43.2 − 24.9i)13-s + (−25.2 + 27.1i)14-s + 16·16-s + (23.2 − 40.3i)17-s + (−14.3 + 8.28i)19-s + (−12.5 + 21.7i)20-s + (−35.7 − 61.9i)22-s + (42.8 + 24.7i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.280 − 0.485i)5-s + (0.732 + 0.680i)7-s − 0.353i·8-s + (0.343 + 0.198i)10-s + (−0.849 + 0.490i)11-s + (0.922 − 0.532i)13-s + (−0.481 + 0.517i)14-s + 0.250·16-s + (0.332 − 0.575i)17-s + (−0.173 + 0.0999i)19-s + (−0.140 + 0.242i)20-s + (−0.346 − 0.600i)22-s + (0.388 + 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.327 - 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.023751347\)
\(L(\frac12)\) \(\approx\) \(2.023751347\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
7 \( 1 + (-13.5 - 12.6i)T \)
good5 \( 1 + (-3.13 + 5.42i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.9 - 17.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-43.2 + 24.9i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-23.2 + 40.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (14.3 - 8.28i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-42.8 - 24.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-161. - 93.4i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 291. iT - 2.97e4T^{2} \)
37 \( 1 + (-198. - 344. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-97.2 - 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (115. - 200. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 68.8T + 1.03e5T^{2} \)
53 \( 1 + (-446. - 257. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 211.T + 2.05e5T^{2} \)
61 \( 1 - 161. iT - 2.26e5T^{2} \)
67 \( 1 + 872.T + 3.00e5T^{2} \)
71 \( 1 - 61.8iT - 3.57e5T^{2} \)
73 \( 1 + (405. + 233. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 972.T + 4.93e5T^{2} \)
83 \( 1 + (-51.1 + 88.5i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-242. - 420. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (329. + 190. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.12138419974669006620211300520, −10.03538311684554192026840909510, −9.080725753149940379585544093954, −8.247767849095030543566713546031, −7.55056795707482736054106438088, −6.16912762422931485227530488593, −5.32607619150576069027471161054, −4.55742965345568223331694193322, −2.82000046412024216962678879828, −1.15847964335827358714321712859, 0.868232277138513879752963146636, 2.25134682686951253135446953723, 3.53578740674528480796179591157, 4.62391755787544392790077405019, 5.82314364758427585323448427772, 6.99893439225185619732009319780, 8.181402059400653591383594269563, 8.872086416680793649326659518786, 10.40926821046050680520664067735, 10.55344929517223488149282220478

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