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Properties

Label 2-1260-28.27-c1-0-6
Degree $2$
Conductor $1260$
Sign $0.533 - 0.846i$
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  + (0.780 − 1.17i)2-s + (−0.780 − 1.84i)4-s − i·5-s + (−2.17 + 1.51i)7-s + (−2.78 − 0.516i)8-s + (−1.17 − 0.780i)10-s + 4.71i·11-s + 2i·13-s + (0.0846 + 3.74i)14-s + (−2.78 + 2.87i)16-s − 1.12i·17-s − 4.71·19-s + (−1.84 + 0.780i)20-s + (5.56 + 3.68i)22-s + 6.41i·23-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)2-s + (−0.390 − 0.920i)4-s − 0.447i·5-s + (−0.821 + 0.570i)7-s + (−0.983 − 0.182i)8-s + (−0.372 − 0.246i)10-s + 1.42i·11-s + 0.554i·13-s + (0.0226 + 0.999i)14-s + (−0.695 + 0.718i)16-s − 0.272i·17-s − 1.08·19-s + (−0.411 + 0.174i)20-s + (1.18 + 0.785i)22-s + 1.33i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.533 - 0.846i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8608629486\)
\(L(\frac12)\) \(\approx\) \(0.8608629486\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.780 + 1.17i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (2.17 - 1.51i)T \)
good11 \( 1 - 4.71iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 1.12iT - 17T^{2} \)
19 \( 1 + 4.71T + 19T^{2} \)
23 \( 1 - 6.41iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 3.39T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 1.12iT - 41T^{2} \)
43 \( 1 - 0.371iT - 43T^{2} \)
47 \( 1 + 5.08T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 2.06T + 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 3.76iT - 67T^{2} \)
71 \( 1 + 7.36iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + 1.32iT - 79T^{2} \)
83 \( 1 + 3.02T + 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 + 1.12iT - 97T^{2} \)
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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

−9.663515660084924027152593714903, −9.428367334281080775292353135775, −8.484623294351115977183485585155, −7.17174204991654632638700511846, −6.34775024986455127300799494533, −5.41834827907706398513433762651, −4.57671053574657593717053616885, −3.74906302593296204103428076947, −2.54848580843282777593179862978, −1.63894497498434354511403627438, 0.28827452098549073983828908392, 2.72774182159020843442189523624, 3.50595772621327673447257542001, 4.34397979057363308758542816005, 5.57304911608438301844901100285, 6.35456523306934113970117451905, 6.77979056713959413719964230685, 7.922624829965287140305359867202, 8.482038690267699397534570828493, 9.385993703006984599002858769288

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