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Properties

Label 2-56e2-28.23-c0-0-1
Degree $2$
Conductor $3136$
Sign $-0.922 + 0.386i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.5i)11-s − 0.999i·15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)23-s − i·27-s + (−0.866 + 0.5i)31-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.866 − 0.5i)47-s + (−0.866 − 0.499i)51-s + (−0.5 − 0.866i)53-s − 0.999i·55-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 + 0.866i)5-s + (−0.866 + 0.5i)11-s − 0.999i·15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.866 − 0.5i)23-s − i·27-s + (−0.866 + 0.5i)31-s + (0.499 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.866 − 0.5i)47-s + (−0.866 − 0.499i)51-s + (−0.5 − 0.866i)53-s − 0.999i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-0.922 + 0.386i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ -0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3382508872\)
\(L(\frac12)\) \(\approx\) \(0.3382508872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.555757057599312091557425416914, −8.278070326220119926768325409862, −7.83765742053185850419649072840, −7.00327172588241272348670063683, −6.21037618565509783879556268516, −5.42369667964247506436624431277, −4.82106732501151092768224842990, −3.78943700279738606984776255654, −3.05697506657995878122940178137, −1.83775973602826538768137172394, 0.24347775471864274651135896288, 1.29167139690519508818261393705, 2.75932656545832648552526362290, 3.74134469896958864516111925136, 4.82386888985987820020917570251, 5.45551684977747681528489214917, 5.96033087163633372418457562213, 7.08084507963054419903708899983, 7.60790592500123581390715431203, 8.354452471728912407530097253098

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