Properties

Label 2-245-1.1-c3-0-9
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  + 3.84·2-s − 8.96·3-s + 6.79·4-s − 5·5-s − 34.4·6-s − 4.64·8-s + 53.4·9-s − 19.2·10-s + 23.1·11-s − 60.9·12-s + 61.0·13-s + 44.8·15-s − 72.1·16-s + 0.688·17-s + 205.·18-s + 63.2·19-s − 33.9·20-s + 88.8·22-s + 124.·23-s + 41.6·24-s + 25·25-s + 234.·26-s − 236.·27-s + 104.·29-s + 172.·30-s + 280.·31-s − 240.·32-s + ⋯
L(s)  = 1  + 1.35·2-s − 1.72·3-s + 0.849·4-s − 0.447·5-s − 2.34·6-s − 0.205·8-s + 1.97·9-s − 0.608·10-s + 0.633·11-s − 1.46·12-s + 1.30·13-s + 0.771·15-s − 1.12·16-s + 0.00981·17-s + 2.69·18-s + 0.763·19-s − 0.379·20-s + 0.861·22-s + 1.12·23-s + 0.353·24-s + 0.200·25-s + 1.77·26-s − 1.68·27-s + 0.667·29-s + 1.04·30-s + 1.62·31-s − 1.32·32-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.931260274\)
\(L(\frac12)\) \(\approx\) \(1.931260274\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.84T + 8T^{2} \)
3 \( 1 + 8.96T + 27T^{2} \)
11 \( 1 - 23.1T + 1.33e3T^{2} \)
13 \( 1 - 61.0T + 2.19e3T^{2} \)
17 \( 1 - 0.688T + 4.91e3T^{2} \)
19 \( 1 - 63.2T + 6.85e3T^{2} \)
23 \( 1 - 124.T + 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 - 280.T + 2.97e4T^{2} \)
37 \( 1 + 263.T + 5.06e4T^{2} \)
41 \( 1 + 243.T + 6.89e4T^{2} \)
43 \( 1 - 172.T + 7.95e4T^{2} \)
47 \( 1 + 107.T + 1.03e5T^{2} \)
53 \( 1 - 44.8T + 1.48e5T^{2} \)
59 \( 1 - 457.T + 2.05e5T^{2} \)
61 \( 1 - 473.T + 2.26e5T^{2} \)
67 \( 1 + 229.T + 3.00e5T^{2} \)
71 \( 1 - 407.T + 3.57e5T^{2} \)
73 \( 1 + 348.T + 3.89e5T^{2} \)
79 \( 1 - 840.T + 4.93e5T^{2} \)
83 \( 1 + 885.T + 5.71e5T^{2} \)
89 \( 1 + 856.T + 7.04e5T^{2} \)
97 \( 1 - 189.T + 9.12e5T^{2} \)
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

−11.67351381089574262678423530911, −11.31091935600495410806580134001, −10.20235518682035447779354740259, −8.707818206922948991514822676170, −6.94454311036232921194851144966, −6.30408163407493739120109002436, −5.35117937631344771954764235488, −4.52324531184787193836306656766, −3.44814758247506059171419828911, −0.948381794672485627324477650540, 0.948381794672485627324477650540, 3.44814758247506059171419828911, 4.52324531184787193836306656766, 5.35117937631344771954764235488, 6.30408163407493739120109002436, 6.94454311036232921194851144966, 8.707818206922948991514822676170, 10.20235518682035447779354740259, 11.31091935600495410806580134001, 11.67351381089574262678423530911

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