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Properties

Label 2-378-21.17-c3-0-17
Degree $2$
Conductor $378$
Sign $0.205 + 0.978i$
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  + (−1.73 + i)2-s + (1.99 − 3.46i)4-s + (−5.82 − 10.0i)5-s + (7.93 − 16.7i)7-s + 7.99i·8-s + (20.1 + 11.6i)10-s + (52.2 + 30.1i)11-s + 2.45i·13-s + (2.98 + 36.9i)14-s + (−8 − 13.8i)16-s + (−38.5 + 66.6i)17-s + (118. − 68.1i)19-s − 46.5·20-s − 120.·22-s + (−59.6 + 34.4i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.520 − 0.902i)5-s + (0.428 − 0.903i)7-s + 0.353i·8-s + (0.637 + 0.368i)10-s + (1.43 + 0.826i)11-s + 0.0524i·13-s + (0.0569 + 0.704i)14-s + (−0.125 − 0.216i)16-s + (−0.549 + 0.951i)17-s + (1.42 − 0.822i)19-s − 0.520·20-s − 1.16·22-s + (−0.540 + 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.205 + 0.978i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.247980986\)
\(L(\frac12)\) \(\approx\) \(1.247980986\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - i)T \)
3 \( 1 \)
7 \( 1 + (-7.93 + 16.7i)T \)
good5 \( 1 + (5.82 + 10.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-52.2 - 30.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 2.45iT - 2.19e3T^{2} \)
17 \( 1 + (38.5 - 66.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-118. + 68.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (59.6 - 34.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 279. iT - 2.43e4T^{2} \)
31 \( 1 + (50.9 + 29.4i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (130. + 225. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 246.T + 6.89e4T^{2} \)
43 \( 1 - 503.T + 7.95e4T^{2} \)
47 \( 1 + (-170. - 294. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-19.3 - 11.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (8.94 - 15.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (339. - 196. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-225. + 389. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 + (422. + 243. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (505. + 874. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 471.T + 5.71e5T^{2} \)
89 \( 1 + (182. + 316. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.58e3iT - 9.12e5T^{2} \)
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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

−10.68312061362016144324036869091, −9.556752213181183590294131704135, −8.967519232585291877436200501706, −7.85477808230152747249544497065, −7.23569245392767560721069297492, −6.09078651645074414123686310840, −4.64570231384662097995851040650, −3.96889843225946292912894542286, −1.70641878315182439871498541968, −0.60013818569847286840384207171, 1.31089567092060886015301516334, 2.85563455421768603610928428864, 3.73046705389189460226147678581, 5.38254704905762319679042680984, 6.62212487477641287397097611420, 7.41915736042230823570970434932, 8.587994582591160897681086673911, 9.166886072364897420240653733267, 10.28238845651661881753912106927, 11.39221099499777053639792243764

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