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Properties

Label 2-740-740.739-c0-0-4
Degree $2$
Conductor $740$
Sign $1$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
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  + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{740} (739, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.770809771\)
\(L(\frac12)\) \(\approx\) \(1.770809771\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
37 \( 1 - T \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 + T )^{2} \)
17 \( ( 1 + T )^{2} \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
41 \( ( 1 - T )^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−10.80821644897585403714162290343, −9.780213099441346580515854375829, −9.026933799111392309276303617302, −7.77567985936988259385724288307, −6.79294902490755101631249068117, −6.06124431521073214267295899939, −5.13488793212626901170948348056, −4.41645981063411314247634537539, −2.72816018386619166937329311600, −2.24576785584387077945318324787, 2.24576785584387077945318324787, 2.72816018386619166937329311600, 4.41645981063411314247634537539, 5.13488793212626901170948348056, 6.06124431521073214267295899939, 6.79294902490755101631249068117, 7.77567985936988259385724288307, 9.026933799111392309276303617302, 9.780213099441346580515854375829, 10.80821644897585403714162290343

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