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Properties

Label 2-378-63.16-c1-0-4
Degree $2$
Conductor $378$
Sign $0.296 - 0.954i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 3·5-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (1.5 + 2.59i)10-s + 3·11-s + (−2.5 − 4.33i)13-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s + (−2.5 + 4.33i)19-s + (−1.49 + 2.59i)20-s + (1.5 + 2.59i)22-s + 3·23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.474 + 0.821i)10-s + 0.904·11-s + (−0.693 − 1.20i)13-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.573 + 0.993i)19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s + 0.625·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.296 - 0.954i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52529 + 1.12326i\)
\(L(\frac12)\) \(\approx\) \(1.52529 + 1.12326i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.79266121519492507808887655519, −10.39886497102759487159366782188, −9.619018142985817774990648318467, −8.818895695303352818051336305808, −7.80083606031369702444328819105, −6.41502720249413499461459530719, −5.86709639487708148459070887951, −5.04012728088693649310951558838, −3.38853552349609335559005879725, −2.01824613382574565182518686881, 1.38123911080229985451394679110, 2.69409037696983255698842367186, 4.18234332028656299378009613305, 5.10149594255908444749909301453, 6.47269556162318140505526488840, 7.01373435114625526197951900274, 8.812875209103618643700518907390, 9.647416055390662117375724183757, 10.11373510764729176653577055372, 11.24405396601319947006869578317

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