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Properties

Label 2-1260-28.27-c1-0-63
Degree $2$
Conductor $1260$
Sign $0.327 + 0.944i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (−1.73 − i)4-s − i·5-s + (1.73 − 2i)7-s + (2 − 1.99i)8-s + (1.36 + 0.366i)10-s + 0.267i·11-s + 0.464i·13-s + (2.09 + 3.09i)14-s + (1.99 + 3.46i)16-s − 6.46i·17-s − 6·19-s + (−1 + 1.73i)20-s + (−0.366 − 0.0980i)22-s + 1.46i·23-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s − 0.447i·5-s + (0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (0.431 + 0.115i)10-s + 0.0807i·11-s + 0.128i·13-s + (0.560 + 0.827i)14-s + (0.499 + 0.866i)16-s − 1.56i·17-s − 1.37·19-s + (−0.223 + 0.387i)20-s + (−0.0780 − 0.0209i)22-s + 0.305i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8828823631\)
\(L(\frac12)\) \(\approx\) \(0.8828823631\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 - 1.36i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (-1.73 + 2i)T \)
good11 \( 1 - 0.267iT - 11T^{2} \)
13 \( 1 - 0.464iT - 13T^{2} \)
17 \( 1 + 6.46iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 1.46iT - 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 9.46T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 1.73T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 9.46iT - 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−9.283056971147389595088087154615, −8.675656947433121645080888479241, −7.77556942715874769224264796817, −7.20159738874756807368829802223, −6.34526710080354409511228400849, −5.21324941422193871365956030584, −4.66516084301488238001165598389, −3.71539924097507702492781206413, −1.83413695762571736137334436232, −0.40730130530226666974879295826, 1.64114303492814768542305209505, 2.44995537663784155786992805513, 3.62542349700719854761917951003, 4.48739208313487072455953898385, 5.55495079547206395839902428998, 6.46729655932502257039330089598, 7.76713451448569319060798413424, 8.423687059274205819144679731561, 8.982364388878318663942184412809, 10.05364300470033144001415607089

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