Properties

Label 2-1260-7.4-c1-0-8
Degree $2$
Conductor $1260$
Sign $0.749 + 0.661i$
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  + (−0.5 − 0.866i)5-s + (1.62 − 2.09i)7-s + (−2.12 + 3.67i)11-s + 3.24·13-s + (2.12 − 3.67i)17-s + (3.5 + 6.06i)19-s + (−2.12 − 3.67i)23-s + (−0.499 + 0.866i)25-s + 1.75·29-s + (4.74 − 8.21i)31-s + (−2.62 − 0.358i)35-s + (−1.62 − 2.80i)37-s + 4.24·41-s + 3.24·43-s + (−3 − 5.19i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.612 − 0.790i)7-s + (−0.639 + 1.10i)11-s + 0.899·13-s + (0.514 − 0.891i)17-s + (0.802 + 1.39i)19-s + (−0.442 − 0.766i)23-s + (−0.0999 + 0.173i)25-s + 0.326·29-s + (0.851 − 1.47i)31-s + (−0.443 − 0.0606i)35-s + (−0.266 − 0.461i)37-s + 0.662·41-s + 0.494·43-s + (−0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.712818290\)
\(L(\frac12)\) \(\approx\) \(1.712818290\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.62 + 2.09i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 + 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.75T + 29T^{2} \)
31 \( 1 + (-4.74 + 8.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.62 + 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.24 + 7.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.62 + 4.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (4.62 - 8.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.485T + 97T^{2} \)
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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

−9.844344911162434351305636934568, −8.659800891764323070644507428374, −7.77516537911387945102871271155, −7.48111374356440582283825032620, −6.24575771342782061880304356322, −5.24305907986460708846884144149, −4.43972873644595143326666936391, −3.61402089681465346180268872021, −2.15007980847801511799241417155, −0.878529131067263648826946604968, 1.22003362621220431944909068351, 2.72897962206069743905976128045, 3.45965757342017302028359926700, 4.77369789118127068714612529079, 5.65521055207275930482434541682, 6.31123394377091446421841536434, 7.46664122250502789298287201201, 8.290521503060328428408954730525, 8.740340814384956135065512696791, 9.770247853907568020374926589454

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