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Properties

Label 2-1260-35.34-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.222033544\)
\(L(\frac12)\) \(\approx\) \(1.222033544\)
\(L(1)\) 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

−9.873693344346044659939045928046, −9.005882923096353191593402405079, −8.661563187286275510799554047031, −7.18883459404355976420529118672, −6.30858447754219623007341995496, −6.11135291301415038936987988681, −4.77218924779916141471915123661, −3.72379200753781848386757226564, −2.70275531771333409161355725429, −1.41314304308891076427101357965, 1.41314304308891076427101357965, 2.70275531771333409161355725429, 3.72379200753781848386757226564, 4.77218924779916141471915123661, 6.11135291301415038936987988681, 6.30858447754219623007341995496, 7.18883459404355976420529118672, 8.661563187286275510799554047031, 9.005882923096353191593402405079, 9.873693344346044659939045928046

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