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Properties

Label 2-351-39.8-c1-0-17
Degree $2$
Conductor $351$
Sign $-0.733 + 0.679i$
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  + (1.94 − 1.94i)2-s − 5.55i·4-s + (0.426 − 0.426i)5-s + (−1.60 + 1.60i)7-s + (−6.90 − 6.90i)8-s − 1.65i·10-s + (−1.16 − 1.16i)11-s + (3.11 + 1.81i)13-s + 6.21i·14-s − 15.7·16-s + 6.73·17-s + (−1.61 − 1.61i)19-s + (−2.37 − 2.37i)20-s − 4.53·22-s + 7.07·23-s + ⋯
L(s)  = 1  + (1.37 − 1.37i)2-s − 2.77i·4-s + (0.190 − 0.190i)5-s + (−0.604 + 0.604i)7-s + (−2.44 − 2.44i)8-s − 0.524i·10-s + (−0.351 − 0.351i)11-s + (0.863 + 0.504i)13-s + 1.66i·14-s − 3.93·16-s + 1.63·17-s + (−0.371 − 0.371i)19-s + (−0.529 − 0.529i)20-s − 0.966·22-s + 1.47·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927726 - 2.36481i\)
\(L(\frac12)\) \(\approx\) \(0.927726 - 2.36481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 - 1.81i)T \)
good2 \( 1 + (-1.94 + 1.94i)T - 2iT^{2} \)
5 \( 1 + (-0.426 + 0.426i)T - 5iT^{2} \)
7 \( 1 + (1.60 - 1.60i)T - 7iT^{2} \)
11 \( 1 + (1.16 + 1.16i)T + 11iT^{2} \)
17 \( 1 - 6.73T + 17T^{2} \)
19 \( 1 + (1.61 + 1.61i)T + 19iT^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 - 4.74iT - 29T^{2} \)
31 \( 1 + (3.99 + 3.99i)T + 31iT^{2} \)
37 \( 1 + (5.84 - 5.84i)T - 37iT^{2} \)
41 \( 1 + (-1.16 + 1.16i)T - 41iT^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + (5.13 + 5.13i)T + 47iT^{2} \)
53 \( 1 + 0.512iT - 53T^{2} \)
59 \( 1 + (-4.62 - 4.62i)T + 59iT^{2} \)
61 \( 1 + 1.50T + 61T^{2} \)
67 \( 1 + (-3.28 - 3.28i)T + 67iT^{2} \)
71 \( 1 + (-7.49 + 7.49i)T - 71iT^{2} \)
73 \( 1 + (4.30 - 4.30i)T - 73iT^{2} \)
79 \( 1 + 9.54T + 79T^{2} \)
83 \( 1 + (1.71 - 1.71i)T - 83iT^{2} \)
89 \( 1 + (5.47 + 5.47i)T + 89iT^{2} \)
97 \( 1 + (-5.52 - 5.52i)T + 97iT^{2} \)
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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

−11.30905349656703249929047462423, −10.56237604185269720100884266617, −9.585104764254061603650789036229, −8.821615680100566653937795568374, −6.82485637536885284023307985992, −5.71179393950514340467950073041, −5.14016979125200991977167048669, −3.66325889437686009998053104737, −2.91737654737040511842422196536, −1.39299378550438797481528785478, 3.04993802272005227316804183195, 3.87054988912702236327645913464, 5.14450479678198077508850472409, 5.98656801246251093243255216671, 6.87931808441396797896901898855, 7.68563089609909401651783201925, 8.572142202922962522251265202522, 9.975487869621972654289302672259, 11.13343306640861001067508216108, 12.47685838036481103185043660532

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