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Properties

Label 2-1260-1260.419-c0-0-3
Degree $2$
Conductor $1260$
Sign $-0.766 + 0.642i$
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  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s − 0.999i·10-s + (−0.866 + 0.499i)12-s − 0.999·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s − 0.999i·10-s + (−0.866 + 0.499i)12-s − 0.999·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.276212739\)
\(L(\frac12)\) \(\approx\) \(1.276212739\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.716647113164145661304169904686, −9.101857948688342680507353097287, −7.58706063454325525774045507735, −6.85234005909318328479620538192, −5.98906037052576786508416166231, −5.38773238138527138026279814434, −4.54937628855373487676906133491, −3.51091774521443874665286719630, −2.10149099611464664625096464124, −0.948050536544041365989089782437, 2.37945671012028539417579573334, 3.37196237862551568776513028710, 4.23298361836153261841330256782, 5.52453209054353937105380945534, 5.81028633936786909154771350224, 6.78021881009993992671403625923, 7.17723765765442203687340013052, 8.606667575155535234664615223402, 9.570281839504335688484285424420, 10.22847689086900245240067040943

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