Properties

Label 2-378-189.83-c1-0-2
Degree $2$
Conductor $378$
Sign $-0.865 - 0.501i$
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  + (−0.984 − 0.173i)2-s + (0.754 + 1.55i)3-s + (0.939 + 0.342i)4-s + (−0.486 + 0.408i)5-s + (−0.472 − 1.66i)6-s + (−2.18 − 1.48i)7-s + (−0.866 − 0.5i)8-s + (−1.86 + 2.35i)9-s + (0.550 − 0.317i)10-s + (−3.72 + 4.43i)11-s + (0.175 + 1.72i)12-s + (0.505 − 0.0891i)13-s + (1.89 + 1.84i)14-s + (−1.00 − 0.450i)15-s + (0.766 + 0.642i)16-s + (1.66 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.122i)2-s + (0.435 + 0.900i)3-s + (0.469 + 0.171i)4-s + (−0.217 + 0.182i)5-s + (−0.192 − 0.680i)6-s + (−0.826 − 0.562i)7-s + (−0.306 − 0.176i)8-s + (−0.620 + 0.784i)9-s + (0.174 − 0.100i)10-s + (−1.12 + 1.33i)11-s + (0.0507 + 0.497i)12-s + (0.140 − 0.0247i)13-s + (0.506 + 0.493i)14-s + (−0.259 − 0.116i)15-s + (0.191 + 0.160i)16-s + (0.404 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149211 + 0.555374i\)
\(L(\frac12)\) \(\approx\) \(0.149211 + 0.555374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (-0.754 - 1.55i)T \)
7 \( 1 + (2.18 + 1.48i)T \)
good5 \( 1 + (0.486 - 0.408i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (3.72 - 4.43i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.505 + 0.0891i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.66 - 2.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.56 + 2.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.488 - 1.34i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (3.07 + 0.541i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.46 + 6.77i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (0.189 + 0.328i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.877 - 4.97i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.02 - 5.89i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (5.23 - 1.90i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 3.85iT - 53T^{2} \)
59 \( 1 + (-2.85 + 2.39i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.871 + 2.39i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.36 + 7.75i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.67 - 1.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.5 - 6.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.238 - 1.35i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.33 + 7.57i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.08 + 3.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.4 - 13.5i)T + (-16.8 - 95.5i)T^{2} \)
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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

−11.31253895365970928597379951252, −10.57387346223830353010715993915, −9.868535939140699986501482691091, −9.309346076374019168826962794169, −7.998458988168337715772537285597, −7.42980242470428183225055349285, −6.08845784605018474943204361192, −4.63795223705386414232150462485, −3.55122996781532694581381853150, −2.34519716176487821414055159290, 0.42517260809805096198502751196, 2.38639766401936764598854497665, 3.37431151770909329781600527398, 5.54600788054892969760214491236, 6.32754506816499534768474368311, 7.36974926227408522083173531600, 8.412928813212897491371878351391, 8.736037361780162030546300486454, 9.956403098389953037913483078626, 10.91960702238197362127835789563

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