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Properties

Label 2-378-63.4-c1-0-0
Degree $2$
Conductor $378$
Sign $-0.384 - 0.923i$
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 − 3.69·5-s + (−1.40 + 2.24i)7-s − 0.999·8-s + (−1.84 + 3.20i)10-s + 1.47·11-s + (−1.34 + 2.33i)13-s + (1.23 + 2.33i)14-s + (−0.5 + 0.866i)16-s + (−3.28 + 5.69i)17-s + (−0.444 − 0.769i)19-s + (1.84 + 3.20i)20-s + (0.738 − 1.27i)22-s − 6.28·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.65·5-s + (−0.531 + 0.847i)7-s − 0.353·8-s + (−0.584 + 1.01i)10-s + 0.445·11-s + (−0.374 + 0.648i)13-s + (0.331 + 0.624i)14-s + (−0.125 + 0.216i)16-s + (−0.797 + 1.38i)17-s + (−0.101 − 0.176i)19-s + (0.413 + 0.716i)20-s + (0.157 − 0.272i)22-s − 1.31·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.384 - 0.923i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176644 + 0.265016i\)
\(L(\frac12)\) \(\approx\) \(0.176644 + 0.265016i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 - 2.24i)T \)
good5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 + (1.34 - 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.444 + 0.769i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.28T + 23T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.40 + 5.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.38 + 2.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.05 + 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 - 6.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.60 + 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.45 - 5.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.86 + 4.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.73 - 8.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + (6.03 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.72 - 9.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.23 + 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.43 - 7.68i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.58 + 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63756469043483530331648021055, −11.13843277666932521664536189027, −9.890412894623869916762191150990, −8.881897089032190985018288300206, −8.103095588703118388048706954031, −6.86987546305376445864445973712, −5.80955431381666998116031309575, −4.26052969464787512768980660732, −3.77386079925759843084940529891, −2.24540187874159578509297673622, 0.18354262588459638198337137142, 3.22108177328684493248796902058, 4.05456007907721597273083467716, 4.98839035718988539228843506352, 6.57772862997252882010510051345, 7.30141824165643325311456520881, 7.974415821032843342033107441566, 9.019696297029729287670606218584, 10.21190423521122778084985605934, 11.28909498426578132684761425009

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