Properties

Label 2-378-189.101-c1-0-12
Degree $2$
Conductor $378$
Sign $0.999 - 0.000404i$
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 + (1.45 − 0.943i)3-s + (−0.173 − 0.984i)4-s + (0.998 − 0.837i)5-s + (−0.211 + 1.71i)6-s + (1.20 + 2.35i)7-s + (0.866 + 0.500i)8-s + (1.22 − 2.74i)9-s + 1.30i·10-s + (0.519 − 0.618i)11-s + (−1.18 − 1.26i)12-s + (−0.540 + 1.48i)13-s + (−2.57 − 0.589i)14-s + (0.660 − 2.15i)15-s + (−0.939 + 0.342i)16-s + 1.47·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (0.838 − 0.544i)3-s + (−0.0868 − 0.492i)4-s + (0.446 − 0.374i)5-s + (−0.0862 + 0.701i)6-s + (0.455 + 0.890i)7-s + (0.306 + 0.176i)8-s + (0.406 − 0.913i)9-s + 0.412i·10-s + (0.156 − 0.186i)11-s + (−0.340 − 0.365i)12-s + (−0.149 + 0.411i)13-s + (−0.689 − 0.157i)14-s + (0.170 − 0.557i)15-s + (−0.234 + 0.0855i)16-s + 0.358·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.999 - 0.000404i$
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: \((2,\ 378,\ (\ :1/2),\ 0.999 - 0.000404i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58562 + 0.000321009i\)
\(L(\frac12)\) \(\approx\) \(1.58562 + 0.000321009i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 + 0.943i)T \)
7 \( 1 + (-1.20 - 2.35i)T \)
good5 \( 1 + (-0.998 + 0.837i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-0.519 + 0.618i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.540 - 1.48i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
19 \( 1 + 0.568iT - 19T^{2} \)
23 \( 1 + (-2.42 + 6.66i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.907 + 2.49i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.88 + 0.509i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.27 - 2.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.20 + 1.53i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.31 - 7.47i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.424 - 2.40i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-7.73 - 4.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (13.2 + 4.81i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-6.78 - 1.19i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.72 - 7.31i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (11.3 - 6.56i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.3 + 5.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.98 + 7.53i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.99 - 2.54i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 1.92T + 89T^{2} \)
97 \( 1 + (15.6 + 2.76i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.44909784763234350643776690717, −10.08364778276924464501805209599, −9.159893258992558649523401236077, −8.653137412745922804764722770682, −7.80491621613632674532400035638, −6.71771850994871969237683793674, −5.79137936558692867299781706394, −4.56422436641373923184831109272, −2.75339873221592971203973740058, −1.48158325888804130174091573297, 1.67245625768392003038949220453, 3.05659784748490184428995567556, 4.04651523669016732843932257870, 5.26770263895909019208334733802, 7.01713523045912014706289691511, 7.79369647452945504496114686388, 8.724929498009798534165876583753, 9.757098461749362719104833929218, 10.29234671750370650861592019834, 11.01183214655090338877745840453

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