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Properties

Label 2-378-7.2-c1-0-0
Degree $2$
Conductor $378$
Sign $-0.0633 - 0.997i$
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 + (−1.82 + 3.15i)5-s − 2.64·7-s − 0.999·8-s + (1.82 + 3.15i)10-s + (1.82 + 3.15i)11-s − 4.64·13-s + (−1.32 + 2.29i)14-s + (−0.5 + 0.866i)16-s + (1.82 + 3.15i)17-s + (−1 + 1.73i)19-s + 3.64·20-s + 3.64·22-s + (−0.645 + 1.11i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.815 + 1.41i)5-s − 0.999·7-s − 0.353·8-s + (0.576 + 0.998i)10-s + (0.549 + 0.951i)11-s − 1.28·13-s + (−0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.442 + 0.765i)17-s + (−0.229 + 0.397i)19-s + 0.815·20-s + 0.777·22-s + (−0.134 + 0.233i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523816 + 0.558111i\)
\(L(\frac12)\) \(\approx\) \(0.523816 + 0.558111i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + (-1.82 - 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.645 - 1.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + (-2.32 - 4.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.96 + 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-2.46 + 4.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.17 - 7.23i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.14 - 1.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15.6T + 71T^{2} \)
73 \( 1 + (5.29 + 9.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.61 - 9.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-2.46 + 4.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−11.85813281806120412180974253395, −10.49251565193596038216178532337, −10.18945282798098072819047880503, −9.144259109859848242965478774415, −7.54679335060942791678207626479, −6.95644672595677303560922918027, −5.89228674916507380439529391223, −4.26986174262211638135468398266, −3.43248451364571585966067193708, −2.35212087482632789877335464207, 0.44371697773959580542864949800, 3.11318357093167785124822727914, 4.30397081887324486776967804390, 5.15945834796982949152461143145, 6.29555167016919467141217585247, 7.38541746279037188231615354084, 8.296726630189243408480826634837, 9.133176837467216471041038355276, 9.883691839286054359074203856205, 11.60665478276366134308843303740

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