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Properties

Label 2-378-189.101-c1-0-2
Degree $2$
Conductor $378$
Sign $-0.994 + 0.102i$
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 + (−0.447 + 1.67i)3-s + (−0.173 − 0.984i)4-s + (1.85 − 1.55i)5-s + (−0.993 − 1.41i)6-s + (−2.64 − 0.142i)7-s + (0.866 + 0.500i)8-s + (−2.59 − 1.49i)9-s + 2.42i·10-s + (−3.21 + 3.83i)11-s + (1.72 + 0.150i)12-s + (−1.16 + 3.18i)13-s + (1.80 − 1.93i)14-s + (1.77 + 3.80i)15-s + (−0.939 + 0.342i)16-s − 6.88·17-s + ⋯
L(s)  = 1  + (−0.454 + 0.541i)2-s + (−0.258 + 0.965i)3-s + (−0.0868 − 0.492i)4-s + (0.829 − 0.696i)5-s + (−0.405 − 0.579i)6-s + (−0.998 − 0.0539i)7-s + (0.306 + 0.176i)8-s + (−0.866 − 0.499i)9-s + 0.765i·10-s + (−0.969 + 1.15i)11-s + (0.498 + 0.0434i)12-s + (−0.321 + 0.884i)13-s + (0.483 − 0.516i)14-s + (0.457 + 0.981i)15-s + (−0.234 + 0.0855i)16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.994 + 0.102i$
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: \((2,\ 378,\ (\ :1/2),\ -0.994 + 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0247636 - 0.481668i\)
\(L(\frac12)\) \(\approx\) \(0.0247636 - 0.481668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.447 - 1.67i)T \)
7 \( 1 + (2.64 + 0.142i)T \)
good5 \( 1 + (-1.85 + 1.55i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (3.21 - 3.83i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.16 - 3.18i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 6.88T + 17T^{2} \)
19 \( 1 - 7.33iT - 19T^{2} \)
23 \( 1 + (-0.798 + 2.19i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0659 + 0.181i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.42 + 0.956i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.33 + 4.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.91 + 2.88i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.69 - 9.62i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.34 + 7.64i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (2.26 + 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.4 - 4.18i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-7.33 - 1.29i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.07 - 6.77i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.60 - 0.927i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.38 + 3.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.10 + 2.60i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.64 - 1.69i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 7.82T + 89T^{2} \)
97 \( 1 + (-5.40 - 0.953i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.74917417940892674863333441053, −10.36566066075551217572643389204, −9.944118529234994377358114713521, −9.260523593120839449139549590297, −8.431363316664739126330098083187, −6.95106908778994044715070095105, −6.07233025010712643690975579262, −5.08306802474250769844735475997, −4.18458732022172847914429620568, −2.24514607727972677953875034844, 0.35315450515745140908606911557, 2.49091504378542210370947171772, 2.96237357054551930456024566253, 5.21968748170458064264382387142, 6.36224793011854426317730453297, 6.93741800931860127603195198281, 8.192628311281253151244495278497, 9.059473279265290004948789796408, 10.20417145490645422681734680483, 10.85387098646057929316060541757

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