Properties

Label 2-378-63.47-c1-0-7
Degree $2$
Conductor $378$
Sign $-0.784 + 0.620i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 3.55·5-s + (−1.49 − 2.18i)7-s − 0.999i·8-s + (−3.07 + 1.77i)10-s − 3.02i·11-s + (0.888 − 0.513i)13-s + (−2.38 − 1.14i)14-s + (−0.5 − 0.866i)16-s + (−0.809 − 1.40i)17-s + (−7.12 − 4.11i)19-s + (−1.77 + 3.07i)20-s + (−1.51 − 2.61i)22-s + 3.35i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 1.58·5-s + (−0.564 − 0.825i)7-s − 0.353i·8-s + (−0.972 + 0.561i)10-s − 0.911i·11-s + (0.246 − 0.142i)13-s + (−0.637 − 0.305i)14-s + (−0.125 − 0.216i)16-s + (−0.196 − 0.339i)17-s + (−1.63 − 0.943i)19-s + (−0.397 + 0.687i)20-s + (−0.322 − 0.558i)22-s + 0.700i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.784 + 0.620i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.784 + 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320020 - 0.920967i\)
\(L(\frac12)\) \(\approx\) \(0.320020 - 0.920967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (1.49 + 2.18i)T \)
good5 \( 1 + 3.55T + 5T^{2} \)
11 \( 1 + 3.02iT - 11T^{2} \)
13 \( 1 + (-0.888 + 0.513i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.809 + 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.12 + 4.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.35iT - 23T^{2} \)
29 \( 1 + (-3.70 - 2.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.18 - 2.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 + 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.57 - 4.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.76 - 1.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.42 + 7.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.06 - 2.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.187 + 0.325i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-1.13 + 0.655i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.462 + 0.800i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.35 - 4.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.3 + 7.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97880860657495310522178796774, −10.62767139578965915721011689624, −9.115440028832886597344925658158, −8.135807969393654573418873055647, −7.13737011435181533014914019157, −6.26851499827229263658647144515, −4.68522444771439530686055974333, −3.89006048082016321932864729506, −2.99436691606120292261955585135, −0.52811519352436816566410284142, 2.55955497085013269397796683488, 3.94506428724070816505568390528, 4.56704383713527680573991798487, 6.10562380810568100910308928409, 6.86590276802874515809502249126, 8.082031506770959716814058519594, 8.534763176791405743266078230931, 9.961117436043775829478262986894, 11.07214794845938712287570206701, 12.06832310598780224149433966337

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