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Properties

Label 2-1568-224.117-c0-0-0
Degree $2$
Conductor $1568$
Sign $0.965 + 0.260i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (0.0999 + 0.758i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (0.292 + 0.707i)22-s + (−0.366 − 1.36i)23-s + (−0.258 + 0.965i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s + (0.707 + 0.707i)36-s + (−1.12 − 1.46i)37-s + (−1.70 − 0.707i)43-s + (0.465 + 0.607i)44-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (0.0999 + 0.758i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (0.292 + 0.707i)22-s + (−0.366 − 1.36i)23-s + (−0.258 + 0.965i)25-s + (0.707 − 0.292i)29-s + (0.258 − 0.965i)32-s + (0.707 + 0.707i)36-s + (−1.12 − 1.46i)37-s + (−1.70 − 0.707i)43-s + (0.465 + 0.607i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.965 + 0.260i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.965 + 0.260i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.054266965\)
\(L(\frac12)\) \(\approx\) \(2.054266965\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (-0.258 - 0.965i)T^{2} \)
5 \( 1 + (0.258 - 0.965i)T^{2} \)
11 \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.12 + 1.46i)T + (-0.258 + 0.965i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T^{2} \)
61 \( 1 + (0.965 + 0.258i)T^{2} \)
67 \( 1 + (1.46 + 1.12i)T + (0.258 + 0.965i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.866 + 0.5i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.891480474460368397718277794111, −8.816204713240631930971826612397, −7.73146554762647488037472533057, −7.10961765238105698113242388152, −6.25200831281817450600271536523, −5.24952137050903959743388463179, −4.62937588043621463287198003580, −3.75830773395126790286800628107, −2.53998353275788361659527029498, −1.70933040767705908283164863611, 1.57330090913512116335388911329, 3.06916107337106384349818976731, 3.65269610480469941550797630183, 4.67072624308762354899598037726, 5.58310076859769727812338605105, 6.41272526544387560706028011272, 6.92518088641719152662339483508, 8.035327656919873058133698767850, 8.633251727966665056032754413653, 9.774585165939241945725329372652

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