Properties

Label 2-245-1.1-c3-0-26
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.31·2-s + 5·3-s + 10.6·4-s + 5·5-s + 21.5·6-s + 11.3·8-s − 2·9-s + 21.5·10-s + 19.7·11-s + 53.1·12-s + 71.3·13-s + 25·15-s − 36.0·16-s − 31.3·17-s − 8.63·18-s + 136.·19-s + 53.1·20-s + 85.1·22-s − 100.·23-s + 56.8·24-s + 25·25-s + 307.·26-s − 145·27-s − 288.·29-s + 107.·30-s − 208.·31-s − 246.·32-s + ⋯
L(s)  = 1  + 1.52·2-s + 0.962·3-s + 1.32·4-s + 0.447·5-s + 1.46·6-s + 0.502·8-s − 0.0740·9-s + 0.682·10-s + 0.540·11-s + 1.27·12-s + 1.52·13-s + 0.430·15-s − 0.562·16-s − 0.447·17-s − 0.113·18-s + 1.64·19-s + 0.594·20-s + 0.825·22-s − 0.914·23-s + 0.483·24-s + 0.200·25-s + 2.32·26-s − 1.03·27-s − 1.84·29-s + 0.656·30-s − 1.21·31-s − 1.36·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.759021610\)
\(L(\frac12)\) \(\approx\) \(5.759021610\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5T \)
7 \( 1 \)
good2 \( 1 - 4.31T + 8T^{2} \)
3 \( 1 - 5T + 27T^{2} \)
11 \( 1 - 19.7T + 1.33e3T^{2} \)
13 \( 1 - 71.3T + 2.19e3T^{2} \)
17 \( 1 + 31.3T + 4.91e3T^{2} \)
19 \( 1 - 136.T + 6.85e3T^{2} \)
23 \( 1 + 100.T + 1.21e4T^{2} \)
29 \( 1 + 288.T + 2.43e4T^{2} \)
31 \( 1 + 208.T + 2.97e4T^{2} \)
37 \( 1 - 309.T + 5.06e4T^{2} \)
41 \( 1 + 181.T + 6.89e4T^{2} \)
43 \( 1 + 18.2T + 7.95e4T^{2} \)
47 \( 1 + 147.T + 1.03e5T^{2} \)
53 \( 1 + 127.T + 1.48e5T^{2} \)
59 \( 1 - 322.T + 2.05e5T^{2} \)
61 \( 1 - 341.T + 2.26e5T^{2} \)
67 \( 1 + 84.3T + 3.00e5T^{2} \)
71 \( 1 + 315.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 - 1.23e3T + 4.93e5T^{2} \)
83 \( 1 + 643.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
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

−11.74587977499222614555230142234, −11.15330591836305629358613737650, −9.540686116183285924535022904666, −8.804817775980628669730559274887, −7.51979363449750788782525662000, −6.19995531418782793662909315371, −5.46802156274272187664260207126, −3.92591194441171380100827390481, −3.27028563145929960322759286153, −1.88561817777381399392277858905, 1.88561817777381399392277858905, 3.27028563145929960322759286153, 3.92591194441171380100827390481, 5.46802156274272187664260207126, 6.19995531418782793662909315371, 7.51979363449750788782525662000, 8.804817775980628669730559274887, 9.540686116183285924535022904666, 11.15330591836305629358613737650, 11.74587977499222614555230142234

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