Properties

Label 2-7360-1.1-c1-0-24
Degree $2$
Conductor $7360$
Sign $1$
Analytic cond. $58.7698$
Root an. cond. $7.66615$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·3-s − 5-s + 1.61·7-s − 2.61·9-s − 3.85·11-s − 4.09·13-s − 0.618·15-s − 5.09·17-s + 4.85·19-s + 1.00·21-s + 23-s + 25-s − 3.47·27-s + 4.76·29-s − 2.09·31-s − 2.38·33-s − 1.61·35-s + 2.47·37-s − 2.52·39-s − 12.3·41-s + 2.61·45-s + 9.70·47-s − 4.38·49-s − 3.14·51-s + 8.47·53-s + 3.85·55-s + 3.00·57-s + ⋯
L(s)  = 1  + 0.356·3-s − 0.447·5-s + 0.611·7-s − 0.872·9-s − 1.16·11-s − 1.13·13-s − 0.159·15-s − 1.23·17-s + 1.11·19-s + 0.218·21-s + 0.208·23-s + 0.200·25-s − 0.668·27-s + 0.884·29-s − 0.375·31-s − 0.414·33-s − 0.273·35-s + 0.406·37-s − 0.404·39-s − 1.92·41-s + 0.390·45-s + 1.41·47-s − 0.625·49-s − 0.440·51-s + 1.16·53-s + 0.519·55-s + 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7360\)    =    \(2^{6} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(58.7698\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273552907\)
\(L(\frac12)\) \(\approx\) \(1.273552907\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - 0.618T + 3T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + 4.09T + 13T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 - 4.85T + 19T^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
31 \( 1 + 2.09T + 31T^{2} \)
37 \( 1 - 2.47T + 37T^{2} \)
41 \( 1 + 12.3T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 9.70T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 + 5.52T + 67T^{2} \)
71 \( 1 - 7.09T + 71T^{2} \)
73 \( 1 + 1.23T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 1.52T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.87721004024048970374644774324, −7.39227872835667151359652745575, −6.68457135669706178736910489784, −5.57915394468678379516364858227, −5.07585733630567753845420230345, −4.47481851651348785179426817894, −3.38924588734296878222409657421, −2.67367340025602793065437752498, −2.05461850907676454989442022701, −0.52666957309215958415525374403, 0.52666957309215958415525374403, 2.05461850907676454989442022701, 2.67367340025602793065437752498, 3.38924588734296878222409657421, 4.47481851651348785179426817894, 5.07585733630567753845420230345, 5.57915394468678379516364858227, 6.68457135669706178736910489784, 7.39227872835667151359652745575, 7.87721004024048970374644774324

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