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Properties

Label 2-1352-104.75-c0-0-2
Degree $2$
Conductor $1352$
Sign $0.711 + 0.702i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.5 + 0.866i)10-s − 0.999·12-s − 0.999·14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.499 − 0.866i)20-s + 0.999·21-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (0.5 + 0.866i)10-s − 0.999·12-s − 0.999·14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.499 − 0.866i)20-s + 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8689575783\)
\(L(\frac12)\) \(\approx\) \(0.8689575783\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.855339659553033433900507536411, −9.037523255936515794198935074887, −8.142546902488715057187245317363, −7.71871737014065492320530361274, −6.74771899739252605207314424248, −4.89295158604076564450827586576, −4.30830815097083046854960840124, −3.60941195263442396701033604260, −2.77379702019684074899418429618, −1.00953677988744670395117215108, 1.33459080947564551581067895464, 2.54358603099472396009383018560, 4.05037507740436950626742637493, 4.99992003657759727172325455197, 6.01705003682056279250632511020, 6.82256053180735567459924108778, 7.81243000993520649585949784565, 8.061293133847610931600101169269, 8.637990824210638062983306887919, 9.639883151340193740955985166158

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