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Properties

Label 2-378-63.4-c1-0-7
Degree $2$
Conductor $378$
Sign $-0.999 - 0.00294i$
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 − 1.76·5-s + (−1.85 − 1.88i)7-s − 0.999·8-s + (−0.880 + 1.52i)10-s − 6.12·11-s + (−0.380 + 0.658i)13-s + (−2.56 + 0.658i)14-s + (−0.5 + 0.866i)16-s + (3.42 − 5.92i)17-s + (0.971 + 1.68i)19-s + (0.880 + 1.52i)20-s + (−3.06 + 5.30i)22-s + 0.421·23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.787·5-s + (−0.699 − 0.714i)7-s − 0.353·8-s + (−0.278 + 0.482i)10-s − 1.84·11-s + (−0.105 + 0.182i)13-s + (−0.684 + 0.176i)14-s + (−0.125 + 0.216i)16-s + (0.829 − 1.43i)17-s + (0.222 + 0.385i)19-s + (0.196 + 0.340i)20-s + (−0.652 + 1.13i)22-s + 0.0877·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.00294i$
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: \((2,\ 378,\ (\ :1/2),\ -0.999 - 0.00294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000903517 + 0.613459i\)
\(L(\frac12)\) \(\approx\) \(0.000903517 + 0.613459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 1.88i)T \)
good5 \( 1 + 1.76T + 5T^{2} \)
11 \( 1 + 6.12T + 11T^{2} \)
13 \( 1 + (0.380 - 0.658i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 + 5.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.971 - 1.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.421T + 23T^{2} \)
29 \( 1 + (0.732 + 1.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.85 + 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.01i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.33 - 7.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.112 + 0.195i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 + 0.265i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.56 - 2.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.30 + 2.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.81 + 3.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.97144798223153818195343124141, −10.11169026628998749026504886155, −9.406912235511257425534819574308, −7.82494211585960394911146107302, −7.40905527821513094616404060027, −5.84556050361431472095173972991, −4.79422875787954971046199739274, −3.64865809708631476687084111966, −2.65340581656796967192014953991, −0.34819613831173027322640924367, 2.73773347524747614503043618865, 3.80189995746138570002847682005, 5.22668999589650630682799591468, 5.89759429932511178320383580650, 7.24046152598763549378944065686, 7.975512004276375213294539089884, 8.774887797352642442662772820154, 10.03046998486854960710988080136, 10.88686479881959868189646208150, 12.13285063752530699729222655532

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