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Properties

Label 2-1260-1260.1147-c0-0-0
Degree $2$
Conductor $1260$
Sign $0.203 + 0.979i$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
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 + (0.707 − 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (−0.258 + 0.965i)7-s + 0.999·8-s − 1.00i·9-s + (0.707 + 0.707i)10-s + (0.866 − 0.5i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (0.965 − 0.258i)14-s + (−0.500 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.707 − 0.707i)3-s + (−0.499 + 0.866i)4-s + (−0.965 + 0.258i)5-s + (−0.965 − 0.258i)6-s + (−0.258 + 0.965i)7-s + 0.999·8-s − 1.00i·9-s + (0.707 + 0.707i)10-s + (0.866 − 0.5i)11-s + (0.258 + 0.965i)12-s + (0.965 − 0.258i)13-s + (0.965 − 0.258i)14-s + (−0.500 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 0.203 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9124862028\)
\(L(\frac12)\) \(\approx\) \(0.9124862028\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
good11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.567237433432570300003375953186, −8.706259701765121751738961877038, −8.384290660514656024378683413879, −7.55810724212145294074289564784, −6.66509403237882870854163402385, −5.57958386770808675404901404909, −3.81627334095731822982170926284, −3.50196319214261299705860292629, −2.45135935143368542288086108791, −1.13191424589445376553934487090, 1.29708298378638582106939934705, 3.33526921864234716435143771297, 4.25016965665686481607256709995, 4.65624388655411653268196830212, 6.08945604046071282957409914592, 7.05921947660831919266869931870, 7.68217021193605930972216487332, 8.437828634573039179718705842854, 9.165609244161786881390500377572, 9.722348352656257875525056198489

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