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Properties

Label 2-1260-1260.223-c0-0-2
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: $\chi_{1260} (223, \cdot )$
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

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

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