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Properties

Label 2-351-39.5-c1-0-12
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: $\chi_{351} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0894813 + 0.228091i\)
\(L(\frac12)\) \(\approx\) \(0.0894813 + 0.228091i\)
\(L(\frac{3}{2})\) 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

−10.58880342052477618436222648903, −10.33316588789297672377204859825, −8.970052987284590326759068806470, −8.619360280880115876887795566948, −7.49188797690061387525299393944, −6.42954376812945883725389763597, −4.16820039747433022104225189377, −3.41122478440646903149660484418, −1.89713152800231567693760973112, −0.25570602557134219938769680629, 1.96263403402319469788486207116, 4.35372398595986271795866069056, 5.86462344614598959101311017343, 6.46733872050131734895027477293, 7.30512658950649443864205201825, 8.446165397455446497721497967646, 9.068717183158546750351790814192, 9.781389266692906200555130699370, 10.84650118275559819173452357274, 11.63906954422249871150105107570

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