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Properties

Label 2-1416-1416.5-c0-0-1
Degree $2$
Conductor $1416$
Sign $0.850 - 0.526i$
Analytic cond. $0.706676$
Root an. cond. $0.840640$
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  + (0.947 + 0.319i)2-s + (0.856 + 0.515i)3-s + (0.796 + 0.605i)4-s + (0.671 − 1.68i)5-s + (0.647 + 0.762i)6-s + (−1.80 + 0.834i)7-s + (0.561 + 0.827i)8-s + (0.468 + 0.883i)9-s + (1.17 − 1.38i)10-s + (0.198 − 0.713i)11-s + (0.370 + 0.928i)12-s + (−1.97 + 0.214i)14-s + (1.44 − 1.09i)15-s + (0.267 + 0.963i)16-s + (0.161 + 0.986i)18-s + ⋯
L(s)  = 1  + (0.947 + 0.319i)2-s + (0.856 + 0.515i)3-s + (0.796 + 0.605i)4-s + (0.671 − 1.68i)5-s + (0.647 + 0.762i)6-s + (−1.80 + 0.834i)7-s + (0.561 + 0.827i)8-s + (0.468 + 0.883i)9-s + (1.17 − 1.38i)10-s + (0.198 − 0.713i)11-s + (0.370 + 0.928i)12-s + (−1.97 + 0.214i)14-s + (1.44 − 1.09i)15-s + (0.267 + 0.963i)16-s + (0.161 + 0.986i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1416\)    =    \(2^{3} \cdot 3 \cdot 59\)
Sign: $0.850 - 0.526i$
Analytic conductor: \(0.706676\)
Root analytic conductor: \(0.840640\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1416} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1416,\ (\ :0),\ 0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.376951101\)
\(L(\frac12)\) \(\approx\) \(2.376951101\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.947 - 0.319i)T \)
3 \( 1 + (-0.856 - 0.515i)T \)
59 \( 1 + (0.647 - 0.762i)T \)
good5 \( 1 + (-0.671 + 1.68i)T + (-0.725 - 0.687i)T^{2} \)
7 \( 1 + (1.80 - 0.834i)T + (0.647 - 0.762i)T^{2} \)
11 \( 1 + (-0.198 + 0.713i)T + (-0.856 - 0.515i)T^{2} \)
13 \( 1 + (0.561 + 0.827i)T^{2} \)
17 \( 1 + (-0.647 - 0.762i)T^{2} \)
19 \( 1 + (-0.907 + 0.419i)T^{2} \)
23 \( 1 + (0.947 + 0.319i)T^{2} \)
29 \( 1 + (0.306 - 0.103i)T + (0.796 - 0.605i)T^{2} \)
31 \( 1 + (1.09 + 0.241i)T + (0.907 + 0.419i)T^{2} \)
37 \( 1 + (0.370 + 0.928i)T^{2} \)
41 \( 1 + (0.947 - 0.319i)T^{2} \)
43 \( 1 + (0.856 - 0.515i)T^{2} \)
47 \( 1 + (0.725 - 0.687i)T^{2} \)
53 \( 1 + (0.606 + 0.714i)T + (-0.161 + 0.986i)T^{2} \)
61 \( 1 + (-0.796 - 0.605i)T^{2} \)
67 \( 1 + (0.370 - 0.928i)T^{2} \)
71 \( 1 + (0.725 - 0.687i)T^{2} \)
73 \( 1 + (1.94 - 0.211i)T + (0.976 - 0.214i)T^{2} \)
79 \( 1 + (0.0927 - 0.0558i)T + (0.468 - 0.883i)T^{2} \)
83 \( 1 + (0.0861 + 1.58i)T + (-0.994 + 0.108i)T^{2} \)
89 \( 1 + (-0.796 + 0.605i)T^{2} \)
97 \( 1 + (0.531 + 0.0578i)T + (0.976 + 0.214i)T^{2} \)
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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

−9.543469366412701270509022538856, −8.939043986596220201405215685794, −8.515433862992388558786615152135, −7.35675951152393924856563921520, −6.13210802702990888086659722354, −5.67813302864861431433763312073, −4.80341075018386132516956074389, −3.79760563140977758313876181594, −3.02340005089068414645663001987, −1.94465829060105661250994572154, 1.84390494431281218405043421207, 2.86598184704426438107232493661, 3.33847434466726770285466224864, 4.12703681312986785702297879937, 5.90768381757035604500102416513, 6.47071548249908751858666151085, 7.12064884664297000750926477805, 7.42388009600039234095329757336, 9.326856085338101928209696796339, 9.808061707561454214911016638473

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