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Properties

Label 2-1260-35.13-c1-0-13
Degree $2$
Conductor $1260$
Sign $-0.502 + 0.864i$
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  + (−1.83 − 1.28i)5-s + (−2.12 + 1.57i)7-s + 4.70·11-s + (1.83 − 1.83i)13-s + (0.737 + 0.737i)17-s − 4.75·19-s + (−3.70 − 3.70i)23-s + (1.70 + 4.70i)25-s + 0.701i·29-s − 8.79i·31-s + (5.91 − 0.160i)35-s + (−3.70 + 3.70i)37-s − 4.75i·41-s + (−5 − 5i)43-s + (−8.05 − 8.05i)47-s + ⋯
L(s)  = 1  + (−0.818 − 0.574i)5-s + (−0.802 + 0.596i)7-s + 1.41·11-s + (0.507 − 0.507i)13-s + (0.178 + 0.178i)17-s − 1.09·19-s + (−0.771 − 0.771i)23-s + (0.340 + 0.940i)25-s + 0.130i·29-s − 1.58i·31-s + (0.999 − 0.0270i)35-s + (−0.608 + 0.608i)37-s − 0.742i·41-s + (−0.762 − 0.762i)43-s + (−1.17 − 1.17i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7579761179\)
\(L(\frac12)\) \(\approx\) \(0.7579761179\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1.83 + 1.28i)T \)
7 \( 1 + (2.12 - 1.57i)T \)
good11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + (-1.83 + 1.83i)T - 13iT^{2} \)
17 \( 1 + (-0.737 - 0.737i)T + 17iT^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + (3.70 + 3.70i)T + 23iT^{2} \)
29 \( 1 - 0.701iT - 29T^{2} \)
31 \( 1 + 8.79iT - 31T^{2} \)
37 \( 1 + (3.70 - 3.70i)T - 37iT^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 + (8.05 + 8.05i)T + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 + 9.50iT - 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 + (-3.11 + 3.11i)T - 73iT^{2} \)
79 \( 1 - 6.70iT - 79T^{2} \)
83 \( 1 + (7.86 - 7.86i)T - 83iT^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + (5.49 + 5.49i)T + 97iT^{2} \)
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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

−9.311083555289766696576910115452, −8.552957769365209308519197767686, −8.064986852732798656587090692989, −6.72797336963340974758556363794, −6.26085933905483155632307885946, −5.15500762485246576473187794591, −4.00133721660409766465527402558, −3.48441565108342843905894699797, −1.94486014066935690554251263957, −0.33198683406876707266111612370, 1.44017245707128326641724401355, 3.12764958687607544589211564798, 3.82544483499963238192736871853, 4.53265864365415813784799598020, 6.19747587014615194461952458886, 6.60652089993961518792538646117, 7.37712409450693252349953323905, 8.343608749891348852089896740828, 9.170261111645590914232022935734, 9.962480165409249745451958533136

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