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Properties

Label 2-1260-28.27-c1-0-20
Degree $2$
Conductor $1260$
Sign $0.0661 - 0.997i$
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  + (−1.10 + 0.887i)2-s + (0.426 − 1.95i)4-s + i·5-s + (−0.391 − 2.61i)7-s + (1.26 + 2.53i)8-s + (−0.887 − 1.10i)10-s − 0.770i·11-s + 5.60i·13-s + (2.75 + 2.53i)14-s + (−3.63 − 1.66i)16-s − 0.503i·17-s − 1.63·19-s + (1.95 + 0.426i)20-s + (0.683 + 0.848i)22-s + 1.42i·23-s + ⋯
L(s)  = 1  + (−0.778 + 0.627i)2-s + (0.213 − 0.977i)4-s + 0.447i·5-s + (−0.148 − 0.988i)7-s + (0.446 + 0.894i)8-s + (−0.280 − 0.348i)10-s − 0.232i·11-s + 1.55i·13-s + (0.735 + 0.677i)14-s + (−0.909 − 0.416i)16-s − 0.122i·17-s − 0.375·19-s + (0.436 + 0.0953i)20-s + (0.145 + 0.180i)22-s + 0.297i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0661 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.0661 - 0.997i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.0661 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9438744544\)
\(L(\frac12)\) \(\approx\) \(0.9438744544\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.887i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (0.391 + 2.61i)T \)
good11 \( 1 + 0.770iT - 11T^{2} \)
13 \( 1 - 5.60iT - 13T^{2} \)
17 \( 1 + 0.503iT - 17T^{2} \)
19 \( 1 + 1.63T + 19T^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 - 8.23T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 - 9.06iT - 43T^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 + 0.455T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 3.32iT - 61T^{2} \)
67 \( 1 - 8.70iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + 2.29iT - 73T^{2} \)
79 \( 1 + 2.56iT - 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 2.98iT - 89T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−9.700378228338708621040478063655, −9.205878413489750413617931228469, −8.096284428211796089087244178595, −7.47339554867651658786236145680, −6.59890758381027059269763465904, −6.19015377693956172482044010855, −4.79719082907147946928063126925, −3.96766548118265862453364967365, −2.45369962033876810413811874945, −1.11679643586909628987565259826, 0.60680449142134696130237027535, 2.12726098354186889570762968682, 2.97006425332859230896274863153, 4.11002938902333089774302156733, 5.29320472100578437785879804941, 6.15577232695206997068675719451, 7.31851194721488602388654761317, 8.214536405483179530741204867682, 8.635925046689846701742241414702, 9.565916474008605340910824164552

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