Properties

Label 2-245-1.1-c5-0-48
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.796·2-s + 10.5·3-s − 31.3·4-s − 25·5-s − 8.42·6-s + 50.4·8-s − 131.·9-s + 19.9·10-s + 525.·11-s − 331.·12-s + 819.·13-s − 264.·15-s + 963.·16-s − 921.·17-s + 104.·18-s − 696.·19-s + 784.·20-s − 418.·22-s + 2.38e3·23-s + 533.·24-s + 625·25-s − 652.·26-s − 3.95e3·27-s − 6.20e3·29-s + 210.·30-s − 8.01e3·31-s − 2.38e3·32-s + ⋯
L(s)  = 1  − 0.140·2-s + 0.678·3-s − 0.980·4-s − 0.447·5-s − 0.0955·6-s + 0.278·8-s − 0.539·9-s + 0.0629·10-s + 1.30·11-s − 0.664·12-s + 1.34·13-s − 0.303·15-s + 0.940·16-s − 0.773·17-s + 0.0760·18-s − 0.442·19-s + 0.438·20-s − 0.184·22-s + 0.938·23-s + 0.189·24-s + 0.200·25-s − 0.189·26-s − 1.04·27-s − 1.36·29-s + 0.0427·30-s − 1.49·31-s − 0.411·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25T \)
7 \( 1 \)
good2 \( 1 + 0.796T + 32T^{2} \)
3 \( 1 - 10.5T + 243T^{2} \)
11 \( 1 - 525.T + 1.61e5T^{2} \)
13 \( 1 - 819.T + 3.71e5T^{2} \)
17 \( 1 + 921.T + 1.41e6T^{2} \)
19 \( 1 + 696.T + 2.47e6T^{2} \)
23 \( 1 - 2.38e3T + 6.43e6T^{2} \)
29 \( 1 + 6.20e3T + 2.05e7T^{2} \)
31 \( 1 + 8.01e3T + 2.86e7T^{2} \)
37 \( 1 + 8.13e3T + 6.93e7T^{2} \)
41 \( 1 - 1.70e3T + 1.15e8T^{2} \)
43 \( 1 + 8.13e3T + 1.47e8T^{2} \)
47 \( 1 + 3.32e3T + 2.29e8T^{2} \)
53 \( 1 + 3.26e4T + 4.18e8T^{2} \)
59 \( 1 - 3.23e4T + 7.14e8T^{2} \)
61 \( 1 - 4.64e4T + 8.44e8T^{2} \)
67 \( 1 + 5.35e3T + 1.35e9T^{2} \)
71 \( 1 + 4.44e4T + 1.80e9T^{2} \)
73 \( 1 + 7.05e4T + 2.07e9T^{2} \)
79 \( 1 - 4.63e4T + 3.07e9T^{2} \)
83 \( 1 - 2.72e4T + 3.93e9T^{2} \)
89 \( 1 + 6.46e4T + 5.58e9T^{2} \)
97 \( 1 - 3.14e4T + 8.58e9T^{2} \)
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

−10.83737086734376389008228315712, −9.274418277148631864371610138734, −8.911377162647097495760981801099, −8.154251433732314968953315595313, −6.83013128016108164559027940572, −5.52312059150619166021419666986, −4.06318157292085137732634478981, −3.44864088181363275860441974354, −1.53698927422100383023646048256, 0, 1.53698927422100383023646048256, 3.44864088181363275860441974354, 4.06318157292085137732634478981, 5.52312059150619166021419666986, 6.83013128016108164559027940572, 8.154251433732314968953315595313, 8.911377162647097495760981801099, 9.274418277148631864371610138734, 10.83737086734376389008228315712

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