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Properties

Label 2-1260-28.27-c1-0-54
Degree $2$
Conductor $1260$
Sign $0.642 + 0.766i$
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  + (−1.28 + 0.599i)2-s + (1.28 − 1.53i)4-s + i·5-s + (2.60 + 0.468i)7-s + (−0.719 + 2.73i)8-s + (−0.599 − 1.28i)10-s − 2.39i·11-s − 2i·13-s + (−3.61 + 0.961i)14-s + (−0.719 − 3.93i)16-s − 7.12i·17-s − 2.39·19-s + (1.53 + 1.28i)20-s + (1.43 + 3.07i)22-s − 5.73i·23-s + ⋯
L(s)  = 1  + (−0.905 + 0.424i)2-s + (0.640 − 0.768i)4-s + 0.447i·5-s + (0.984 + 0.176i)7-s + (−0.254 + 0.967i)8-s + (−0.189 − 0.405i)10-s − 0.723i·11-s − 0.554i·13-s + (−0.966 + 0.257i)14-s + (−0.179 − 0.983i)16-s − 1.72i·17-s − 0.550·19-s + (0.343 + 0.286i)20-s + (0.306 + 0.654i)22-s − 1.19i·23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.642 + 0.766i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9903588038\)
\(L(\frac12)\) \(\approx\) \(0.9903588038\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 0.599i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (-2.60 - 0.468i)T \)
good11 \( 1 + 2.39iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 7.12iT - 17T^{2} \)
19 \( 1 + 2.39T + 19T^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 + 7.60iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 14.2iT - 67T^{2} \)
71 \( 1 + 6.14iT - 71T^{2} \)
73 \( 1 - 9.36iT - 73T^{2} \)
79 \( 1 + 4.27iT - 79T^{2} \)
83 \( 1 - 0.936T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + 7.12iT - 97T^{2} \)
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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

−9.483452041938287608949058392912, −8.618427847961993899709501605987, −8.062739908263648278905089801138, −7.23526779078032219103980602310, −6.44877881315233189446050924775, −5.48441658934365747086502065589, −4.72641207547298598865159042645, −3.07325064906547227328779204667, −2.08686119532292454136582587625, −0.57735473070681414903071578754, 1.43768288994629461067865707136, 2.06005231889101507227453530606, 3.70827344333859387370691369277, 4.44326714572175360507096138271, 5.66113857319878724878749803063, 6.76720130271211091069582702692, 7.61782058330699835380722341593, 8.291768132285506016599650248152, 8.938630278065214406991157016277, 9.804606759914330088187043822248

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