Properties

Label 2-7360-1.1-c1-0-62
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  + 1.56·3-s − 5-s + 2.56·7-s − 0.561·9-s − 2·11-s + 3.56·13-s − 1.56·15-s + 2.56·17-s − 6·19-s + 4·21-s + 23-s + 25-s − 5.56·27-s − 6.12·29-s + 7.24·31-s − 3.12·33-s − 2.56·35-s + 4.56·37-s + 5.56·39-s + 4.12·41-s + 0.561·45-s + 4.68·47-s − 0.438·49-s + 4·51-s + 4.56·53-s + 2·55-s − 9.36·57-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.447·5-s + 0.968·7-s − 0.187·9-s − 0.603·11-s + 0.987·13-s − 0.403·15-s + 0.621·17-s − 1.37·19-s + 0.872·21-s + 0.208·23-s + 0.200·25-s − 1.07·27-s − 1.13·29-s + 1.30·31-s − 0.543·33-s − 0.432·35-s + 0.749·37-s + 0.890·39-s + 0.643·41-s + 0.0837·45-s + 0.683·47-s − 0.0626·49-s + 0.560·51-s + 0.626·53-s + 0.269·55-s − 1.24·57-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: $\chi_{7360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.783586443\)
\(L(\frac12)\) \(\approx\) \(2.783586443\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 - 2.56T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
29 \( 1 + 6.12T + 29T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 - 4.56T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 - 3.68T + 59T^{2} \)
61 \( 1 - 7.12T + 61T^{2} \)
67 \( 1 - 8.56T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 4.43T + 73T^{2} \)
79 \( 1 - 4.87T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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

−8.096683657834363950212778636406, −7.53646783731309843545785946743, −6.55149312304631036741604650053, −5.77083966711572480116835402839, −5.03750969434301862917920422395, −4.13561623935012745516021747789, −3.61407497452552049798978077256, −2.63418107170488340334633108605, −1.99846768591812328884215442760, −0.806080642201332608709624620064, 0.806080642201332608709624620064, 1.99846768591812328884215442760, 2.63418107170488340334633108605, 3.61407497452552049798978077256, 4.13561623935012745516021747789, 5.03750969434301862917920422395, 5.77083966711572480116835402839, 6.55149312304631036741604650053, 7.53646783731309843545785946743, 8.096683657834363950212778636406

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