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Properties

Label 2-351-9.4-c1-0-2
Degree $2$
Conductor $351$
Sign $-0.989 - 0.144i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (1.10 + 1.91i)2-s + (−1.43 + 2.48i)4-s + (−1.80 + 3.12i)5-s + (0.540 + 0.935i)7-s − 1.90·8-s − 7.95·10-s + (−2.14 − 3.71i)11-s + (0.5 − 0.866i)13-s + (−1.19 + 2.06i)14-s + (0.762 + 1.32i)16-s + 2.31·17-s + 1.50·19-s + (−5.16 − 8.94i)20-s + (4.72 − 8.19i)22-s + (−3.54 + 6.13i)23-s + ⋯
L(s)  = 1  + (0.779 + 1.35i)2-s + (−0.716 + 1.24i)4-s + (−0.806 + 1.39i)5-s + (0.204 + 0.353i)7-s − 0.673·8-s − 2.51·10-s + (−0.646 − 1.11i)11-s + (0.138 − 0.240i)13-s + (−0.318 + 0.551i)14-s + (0.190 + 0.330i)16-s + 0.562·17-s + 0.344·19-s + (−1.15 − 2.00i)20-s + (1.00 − 1.74i)22-s + (−0.738 + 1.27i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.989 - 0.144i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.989 - 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120225 + 1.65057i\)
\(L(\frac12)\) \(\approx\) \(0.120225 + 1.65057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1.10 - 1.91i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.80 - 3.12i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.540 - 0.935i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.14 + 3.71i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.31T + 17T^{2} \)
19 \( 1 - 1.50T + 19T^{2} \)
23 \( 1 + (3.54 - 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.86 - 4.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.22 - 2.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.00T + 37T^{2} \)
41 \( 1 + (-5.29 + 9.16i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 6.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.992 + 1.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (-1.84 + 3.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.840 - 1.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.46 + 4.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.07T + 71T^{2} \)
73 \( 1 + 3.54T + 73T^{2} \)
79 \( 1 + (2.03 + 3.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.31 + 12.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.03T + 89T^{2} \)
97 \( 1 + (-0.0682 - 0.118i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.94697055856703445263007205287, −11.09406060855394867646238224591, −10.27342995967665285089905030410, −8.621775515541380092183385814280, −7.72934296460383360236998158002, −7.21852580144919203847707428729, −6.05493493636775160291753627446, −5.40649433605397650427763381884, −3.88395943743011981692943340894, −3.02622584288962884317655210907, 0.971629584716324693052504841184, 2.47820474559038388793995970140, 4.22701188614519312836912640118, 4.41454994164041735153562732972, 5.58218356691221934686639589894, 7.47408177665052256254819807689, 8.272989682538362006500705337272, 9.551111250965302897811337959545, 10.28722136373538654955734376531, 11.36512248926972433442147590712

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