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Properties

Label 2-1260-35.34-c0-0-0
Degree $2$
Conductor $1260$
Sign $1$
Analytic cond. $0.628821$
Root an. cond. $0.792982$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s + 11-s − 13-s + 17-s + 25-s + 29-s − 35-s + 47-s + 49-s − 55-s + 65-s − 2·71-s + 2·73-s + 77-s − 79-s − 2·83-s − 85-s − 91-s − 97-s − 103-s − 109-s + 119-s + ⋯
L(s)  = 1  − 5-s + 7-s + 11-s − 13-s + 17-s + 25-s + 29-s − 35-s + 47-s + 49-s − 55-s + 65-s − 2·71-s + 2·73-s + 77-s − 79-s − 2·83-s − 85-s − 91-s − 97-s − 103-s − 109-s + 119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.628821\)
Root analytic conductor: \(0.792982\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1260} (1189, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046871039\)
\(L(\frac12)\) \(\approx\) \(1.046871039\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 - T \)
good11 \( 1 - T + T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 - T + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−9.911301653987530317790273183499, −8.940115085253413437242662560951, −8.195368340067852777645763548670, −7.50468505971011468070945463499, −6.81511930729578180791902944322, −5.56114605695149092196146774800, −4.64589858273141301680006647496, −3.95066294080835293551130471951, −2.76733555406852481503143319556, −1.25835423753378328678959688166, 1.25835423753378328678959688166, 2.76733555406852481503143319556, 3.95066294080835293551130471951, 4.64589858273141301680006647496, 5.56114605695149092196146774800, 6.81511930729578180791902944322, 7.50468505971011468070945463499, 8.195368340067852777645763548670, 8.940115085253413437242662560951, 9.911301653987530317790273183499

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