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Properties

Label 2-378-9.7-c3-0-16
Degree $2$
Conductor $378$
Sign $-0.683 - 0.730i$
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 + (−9.75 − 16.8i)5-s + (3.5 − 6.06i)7-s − 7.99·8-s − 39.0·10-s + (18.9 − 32.8i)11-s + (−5.34 − 9.24i)13-s + (−7 − 12.1i)14-s + (−8 + 13.8i)16-s − 56.7·17-s + 37.5·19-s + (−39.0 + 67.5i)20-s + (−37.8 − 65.6i)22-s + (−31.1 − 53.9i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.872 − 1.51i)5-s + (0.188 − 0.327i)7-s − 0.353·8-s − 1.23·10-s + (0.519 − 0.899i)11-s + (−0.113 − 0.197i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.809·17-s + 0.453·19-s + (−0.436 + 0.755i)20-s + (−0.367 − 0.635i)22-s + (−0.282 − 0.488i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.683 - 0.730i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.683 - 0.730i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.072558378\)
\(L(\frac12)\) \(\approx\) \(1.072558378\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 + 6.06i)T \)
good5 \( 1 + (9.75 + 16.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18.9 + 32.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (5.34 + 9.24i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 56.7T + 4.91e3T^{2} \)
19 \( 1 - 37.5T + 6.85e3T^{2} \)
23 \( 1 + (31.1 + 53.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-28.8 + 49.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-115. - 200. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 368.T + 5.06e4T^{2} \)
41 \( 1 + (-250. - 434. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-187. + 324. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (280. - 485. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 155.T + 1.48e5T^{2} \)
59 \( 1 + (175. + 304. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-203. + 352. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (383. + 663. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 268.T + 3.57e5T^{2} \)
73 \( 1 + 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + (-120. + 209. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-251. + 434. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-519. + 900. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.59751764598195451754966196710, −9.310047553912485227709589367600, −8.629352013520574824781896311311, −7.81652292037784548513750501027, −6.33621702952691618046492158213, −5.02870734771499673038496176098, −4.36736047393795921227973070883, −3.29291408924588116341535042667, −1.39535468258558520289664800205, −0.34164719674664552765800784340, 2.34307088485359895080241297216, 3.60122729059819026403518602558, 4.51271095943323720170157622103, 5.95170126977055048487812105136, 6.99728315730700839470187163386, 7.38347092052859811600400788859, 8.530552173876305690657085975830, 9.682164848500625981361982220448, 10.72718579714891995728418358350, 11.66753085073373177913278533473

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