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Properties

Label 2-378-7.4-c3-0-17
Degree $2$
Conductor $378$
Sign $0.905 + 0.423i$
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 + (8.05 + 13.9i)5-s + (6.77 − 17.2i)7-s + 7.99·8-s + (16.1 − 27.9i)10-s + (6.04 − 10.4i)11-s − 39.7·13-s + (−36.6 + 5.50i)14-s + (−8 − 13.8i)16-s + (62.7 − 108. i)17-s + (62.5 + 108. i)19-s − 64.4·20-s − 24.1·22-s + (−24.1 − 41.7i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.720 + 1.24i)5-s + (0.365 − 0.930i)7-s + 0.353·8-s + (0.509 − 0.882i)10-s + (0.165 − 0.286i)11-s − 0.848·13-s + (−0.699 + 0.105i)14-s + (−0.125 − 0.216i)16-s + (0.895 − 1.55i)17-s + (0.754 + 1.30i)19-s − 0.720·20-s − 0.234·22-s + (−0.218 − 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.905 + 0.423i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.905 + 0.423i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.853378751\)
\(L(\frac12)\) \(\approx\) \(1.853378751\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.77 + 17.2i)T \)
good5 \( 1 + (-8.05 - 13.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6.04 + 10.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 39.7T + 2.19e3T^{2} \)
17 \( 1 + (-62.7 + 108. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62.5 - 108. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (24.1 + 41.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 93.0T + 2.43e4T^{2} \)
31 \( 1 + (11.3 - 19.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-200. - 347. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 354.T + 6.89e4T^{2} \)
43 \( 1 - 225.T + 7.95e4T^{2} \)
47 \( 1 + (-124. - 214. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-167. + 290. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (91.4 - 158. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (304. + 528. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (59.1 - 102. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 81.4T + 3.57e5T^{2} \)
73 \( 1 + (-489. + 847. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-346. - 600. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + (605. + 1.04e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−10.74348027361018586293532538149, −9.981957764658726910954219956014, −9.543149075560018963042175637749, −7.88575732056653293071748732343, −7.31326885091457613251427849472, −6.19289972475537369919893510651, −4.84462144610333301809115387917, −3.42070518334339030680791431052, −2.50138204065484523785146700870, −0.983860103837046622524081920467, 1.01795088789252218560507228784, 2.28718808642070899351304118311, 4.40290535774035179804260385174, 5.40208925868790961928932729143, 5.92563502899430324353875622607, 7.40361094995492663651709936334, 8.315327578837395807683404661342, 9.226304518652427770438527534104, 9.599120739767576975293783427913, 10.86451169324963758870893628637

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