Properties

Label 2-378-189.59-c1-0-19
Degree $2$
Conductor $378$
Sign $-0.771 + 0.636i$
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  + (0.342 − 0.939i)2-s + (0.320 − 1.70i)3-s + (−0.766 − 0.642i)4-s + (0.0223 + 0.0187i)5-s + (−1.48 − 0.883i)6-s + (2.61 + 0.416i)7-s + (−0.866 + 0.500i)8-s + (−2.79 − 1.09i)9-s + (0.0252 − 0.0145i)10-s + (−3.83 − 4.56i)11-s + (−1.33 + 1.09i)12-s + (2.43 − 2.89i)13-s + (1.28 − 2.31i)14-s + (0.0390 − 0.0320i)15-s + (0.173 + 0.984i)16-s + (0.588 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.241 − 0.664i)2-s + (0.185 − 0.982i)3-s + (−0.383 − 0.321i)4-s + (0.00999 + 0.00838i)5-s + (−0.608 − 0.360i)6-s + (0.987 + 0.157i)7-s + (−0.306 + 0.176i)8-s + (−0.931 − 0.364i)9-s + (0.00798 − 0.00461i)10-s + (−1.15 − 1.37i)11-s + (−0.386 + 0.316i)12-s + (0.674 − 0.804i)13-s + (0.343 − 0.618i)14-s + (0.0100 − 0.00826i)15-s + (0.0434 + 0.246i)16-s + (0.142 + 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.771 + 0.636i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.771 + 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517240 - 1.44026i\)
\(L(\frac12)\) \(\approx\) \(0.517240 - 1.44026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (-0.320 + 1.70i)T \)
7 \( 1 + (-2.61 - 0.416i)T \)
good5 \( 1 + (-0.0223 - 0.0187i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (3.83 + 4.56i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.43 + 2.89i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.588 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.921 + 0.532i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.94 - 8.10i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.310 - 0.369i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (4.43 - 5.28i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 - 8.83T + 37T^{2} \)
41 \( 1 + (9.35 + 7.85i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-10.6 - 3.86i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-4.80 + 4.02i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-7.51 - 4.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.26 + 7.17i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-6.28 - 7.49i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-0.264 + 0.0963i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.14 + 2.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.87iT - 73T^{2} \)
79 \( 1 + (-0.344 - 0.125i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.45 - 3.73i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (1.89 - 3.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.11 + 8.56i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.08565734475635998888018623694, −10.52955155239681365155389139080, −8.900802551357935297194833846753, −8.274642401634852156916625897068, −7.47680806187811822784474411452, −5.86215323075681108940992255995, −5.34963719845246959727031804275, −3.54841695883511299713737865366, −2.46891331492699798613607598256, −1.00238049900182396757982857556, 2.41649102303567823642462763791, 4.14673372950259741817112814950, 4.74185673127217570635877616422, 5.69386802185931229552348203807, 7.13084241074761513519994004085, 8.013681774401671793060524567891, 8.875542975949660957985388102278, 9.827699256674672624417869472695, 10.76240269883875034565160923450, 11.53633934691606329030304315304

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