Properties

Label 2-1260-28.27-c1-0-4
Degree $2$
Conductor $1260$
Sign $-0.914 + 0.404i$
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  + (0.947 + 1.04i)2-s + (−0.204 + 1.98i)4-s i·5-s + (−2.29 + 1.31i)7-s + (−2.28 + 1.66i)8-s + (1.04 − 0.947i)10-s + 0.477i·11-s − 2.96i·13-s + (−3.55 − 1.16i)14-s + (−3.91 − 0.814i)16-s + 3.83i·17-s − 5.31·19-s + (1.98 + 0.204i)20-s + (−0.500 + 0.452i)22-s + 7.60i·23-s + ⋯
L(s)  = 1  + (0.669 + 0.742i)2-s + (−0.102 + 0.994i)4-s − 0.447i·5-s + (−0.868 + 0.496i)7-s + (−0.807 + 0.590i)8-s + (0.332 − 0.299i)10-s + 0.143i·11-s − 0.821i·13-s + (−0.950 − 0.311i)14-s + (−0.979 − 0.203i)16-s + 0.929i·17-s − 1.21·19-s + (0.444 + 0.0457i)20-s + (−0.106 + 0.0963i)22-s + 1.58i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.914 + 0.404i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.914 + 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7951188969\)
\(L(\frac12)\) \(\approx\) \(0.7951188969\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.947 - 1.04i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (2.29 - 1.31i)T \)
good11 \( 1 - 0.477iT - 11T^{2} \)
13 \( 1 + 2.96iT - 13T^{2} \)
17 \( 1 - 3.83iT - 17T^{2} \)
19 \( 1 + 5.31T + 19T^{2} \)
23 \( 1 - 7.60iT - 23T^{2} \)
29 \( 1 + 6.17T + 29T^{2} \)
31 \( 1 + 3.38T + 31T^{2} \)
37 \( 1 + 8.62T + 37T^{2} \)
41 \( 1 + 1.01iT - 41T^{2} \)
43 \( 1 + 6.85iT - 43T^{2} \)
47 \( 1 - 6.21T + 47T^{2} \)
53 \( 1 - 9.30T + 53T^{2} \)
59 \( 1 + 4.88T + 59T^{2} \)
61 \( 1 - 4.75iT - 61T^{2} \)
67 \( 1 - 1.30iT - 67T^{2} \)
71 \( 1 - 9.18iT - 71T^{2} \)
73 \( 1 + 4.49iT - 73T^{2} \)
79 \( 1 - 8.80iT - 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 1.60iT - 97T^{2} \)
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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

−10.03756342207474253860491441397, −9.020216467839851276066571898969, −8.551287087715822835126893092955, −7.55061256560996120034474078585, −6.80686637309102907690515929106, −5.65538907793208255988719815845, −5.53098267087325937753597736534, −4.06085877665844860221397878956, −3.44116634504635294245187671578, −2.12547137246431547586054220519, 0.24435487051183644983169442760, 2.00051272489352150894579984350, 2.98636156482953676338930798073, 3.92524687926565706982665119524, 4.67311337712612272963180218290, 5.89987890301830376852792059756, 6.62447595216805361956624619085, 7.24867130600110794040237805308, 8.756901264961865497952206857667, 9.385685785436594357381051156088

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