Properties

Label 2-140-35.34-c0-0-0
Degree $2$
Conductor $140$
Sign $1$
Analytic cond. $0.0698691$
Root an. cond. $0.264327$
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  − 3-s + 5-s + 7-s − 11-s − 13-s − 15-s − 17-s − 21-s + 25-s + 27-s − 29-s + 33-s + 35-s + 39-s − 47-s + 49-s + 51-s − 55-s − 65-s + 2·71-s + 2·73-s − 75-s − 77-s − 79-s − 81-s + 2·83-s − 85-s + ⋯
L(s)  = 1  − 3-s + 5-s + 7-s − 11-s − 13-s − 15-s − 17-s − 21-s + 25-s + 27-s − 29-s + 33-s + 35-s + 39-s − 47-s + 49-s + 51-s − 55-s − 65-s + 2·71-s + 2·73-s − 75-s − 77-s − 79-s − 81-s + 2·83-s − 85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.0698691\)
Root analytic conductor: \(0.264327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{140} (69, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5430031738\)
\(L(\frac12)\) \(\approx\) \(0.5430031738\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
11 \( 1 + T + T^{2} \)
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

−13.33065956965597523521545971664, −12.34723408478985616625858398180, −11.21644479547259758976655771914, −10.60373722898220520172273157655, −9.436856976615496154332704613158, −8.102245190597348926961755827512, −6.75170062241563553981571022546, −5.48530009143023484215244134133, −4.87037486212895093408596806754, −2.27790387327828597147359808295, 2.27790387327828597147359808295, 4.87037486212895093408596806754, 5.48530009143023484215244134133, 6.75170062241563553981571022546, 8.102245190597348926961755827512, 9.436856976615496154332704613158, 10.60373722898220520172273157655, 11.21644479547259758976655771914, 12.34723408478985616625858398180, 13.33065956965597523521545971664

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