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Properties

Label 2-378-7.4-c3-0-30
Degree $2$
Conductor $378$
Sign $-0.836 - 0.548i$
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 + (−1.18 − 2.05i)5-s + (9.15 − 16.0i)7-s + 7.99·8-s + (−2.37 + 4.11i)10-s + (−19.5 + 33.8i)11-s − 24.4·13-s + (−37.0 + 0.232i)14-s + (−8 − 13.8i)16-s + (23.8 − 41.3i)17-s + (−59.8 − 103. i)19-s + 9.49·20-s + 78.2·22-s + (76.6 + 132. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.106 − 0.183i)5-s + (0.494 − 0.869i)7-s + 0.353·8-s + (−0.0750 + 0.130i)10-s + (−0.535 + 0.928i)11-s − 0.521·13-s + (−0.707 + 0.00444i)14-s + (−0.125 − 0.216i)16-s + (0.340 − 0.589i)17-s + (−0.722 − 1.25i)19-s + 0.106·20-s + 0.757·22-s + (0.694 + 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.836 - 0.548i$
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: \((2,\ 378,\ (\ :3/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2900862150\)
\(L(\frac12)\) \(\approx\) \(0.2900862150\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.15 + 16.0i)T \)
good5 \( 1 + (1.18 + 2.05i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (19.5 - 33.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 24.4T + 2.19e3T^{2} \)
17 \( 1 + (-23.8 + 41.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (59.8 + 103. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76.6 - 132. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 215.T + 2.43e4T^{2} \)
31 \( 1 + (-29.6 + 51.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-104. - 181. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 415.T + 6.89e4T^{2} \)
43 \( 1 + 452.T + 7.95e4T^{2} \)
47 \( 1 + (114. + 198. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (220. - 382. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (362. - 627. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-170. - 295. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (125. - 217. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 209.T + 3.57e5T^{2} \)
73 \( 1 + (60.9 - 105. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (399. + 692. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 116.T + 5.71e5T^{2} \)
89 \( 1 + (183. + 317. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−10.34771758917671083178125895112, −9.686295371886940139355532077779, −8.630560553461050459053244800167, −7.56458535092985428199408955348, −6.97570105588297261966166993615, −5.08973377088046101375225602635, −4.40341880655783010650529864918, −2.95686637985386981045000805738, −1.59864834948383628067504249978, −0.10912641217663954273878622179, 1.81026008467172300889308270489, 3.34004953214224646324229385817, 4.94001701198217191779323394798, 5.75273510563762116970292782734, 6.72808616617112336084934256332, 8.034260664968598935907755315314, 8.428234716298307878510645783733, 9.502816590193674645013066008295, 10.57329899543186864987066672785, 11.27090758273602956313623882101

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