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Properties

Label 2-378-189.101-c1-0-9
Degree $2$
Conductor $378$
Sign $0.986 - 0.165i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)2-s + (−1.02 + 1.39i)3-s + (−0.173 − 0.984i)4-s + (0.192 − 0.161i)5-s + (−0.405 − 1.68i)6-s + (1.12 − 2.39i)7-s + (0.866 + 0.500i)8-s + (−0.882 − 2.86i)9-s + 0.250i·10-s + (0.990 − 1.17i)11-s + (1.55 + 0.771i)12-s + (0.561 − 1.54i)13-s + (1.11 + 2.39i)14-s + (0.0269 + 0.433i)15-s + (−0.939 + 0.342i)16-s + 3.55·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.594 + 0.804i)3-s + (−0.0868 − 0.492i)4-s + (0.0859 − 0.0721i)5-s + (−0.165 − 0.687i)6-s + (0.423 − 0.905i)7-s + (0.306 + 0.176i)8-s + (−0.294 − 0.955i)9-s + 0.0793i·10-s + (0.298 − 0.355i)11-s + (0.447 + 0.222i)12-s + (0.155 − 0.428i)13-s + (0.298 + 0.641i)14-s + (0.00695 + 0.111i)15-s + (−0.234 + 0.0855i)16-s + 0.861·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906639 + 0.0754829i\)
\(L(\frac12)\) \(\approx\) \(0.906639 + 0.0754829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 - 0.766i)T \)
3 \( 1 + (1.02 - 1.39i)T \)
7 \( 1 + (-1.12 + 2.39i)T \)
good5 \( 1 + (-0.192 + 0.161i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.990 + 1.17i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.561 + 1.54i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + 7.87iT - 19T^{2} \)
23 \( 1 + (0.889 - 2.44i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-2.08 - 5.72i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.24 + 0.924i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.66 + 2.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.882 + 0.321i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-0.314 + 1.78i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.20 + 6.83i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-8.07 - 4.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.02 + 2.55i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (7.43 + 1.31i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.29 - 1.08i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-9.53 + 5.50i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.9 + 6.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.72 + 8.16i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.82 - 3.21i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 6.25T + 89T^{2} \)
97 \( 1 + (-14.9 - 2.64i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.06402841432099076046240685634, −10.50907798645782566011806854635, −9.549445307974504264423421918817, −8.762633706747123550318430808904, −7.57899486474142355666019325330, −6.66232990687117304962388337135, −5.51558309409333650981292074556, −4.69637394303757397303175141991, −3.42890013347028970182969581311, −0.885342006314742118053960005556, 1.43777020898731211948086225360, 2.57400600657829695013458111099, 4.35517851570279971492404251088, 5.71856171502953226622594582802, 6.51344913622556268845807578551, 7.909813576247392203620549320084, 8.329816493082663690744386865134, 9.695079726136768947395859777249, 10.45119284109731626188564941795, 11.61763863645556598762513241525

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