Properties

Label 2-7e2-1.1-c25-0-33
Degree $2$
Conductor $49$
Sign $1$
Analytic cond. $194.038$
Root an. cond. $13.9297$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.49e3·2-s + 5.76e5·3-s + 2.26e7·4-s − 4.14e8·5-s + 4.32e9·6-s − 8.15e10·8-s − 5.15e11·9-s − 3.10e12·10-s + 7.00e12·11-s + 1.30e13·12-s + 1.39e13·13-s − 2.38e14·15-s − 1.37e15·16-s + 2.13e15·17-s − 3.86e15·18-s + 9.59e15·19-s − 9.39e15·20-s + 5.25e16·22-s − 5.44e16·23-s − 4.69e16·24-s − 1.26e17·25-s + 1.04e17·26-s − 7.85e17·27-s − 1.99e18·29-s − 1.78e18·30-s + 4.95e18·31-s − 7.55e18·32-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.625·3-s + 0.676·4-s − 0.758·5-s + 0.810·6-s − 0.419·8-s − 0.608·9-s − 0.982·10-s + 0.673·11-s + 0.423·12-s + 0.166·13-s − 0.474·15-s − 1.21·16-s + 0.887·17-s − 0.787·18-s + 0.994·19-s − 0.512·20-s + 0.871·22-s − 0.518·23-s − 0.262·24-s − 0.424·25-s + 0.215·26-s − 1.00·27-s − 1.04·29-s − 0.614·30-s + 1.12·31-s − 1.15·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $1$
Analytic conductor: \(194.038\)
Root analytic conductor: \(13.9297\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(4.340265632\)
\(L(\frac12)\) \(\approx\) \(4.340265632\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - 7.49e3T + 3.35e7T^{2} \)
3 \( 1 - 5.76e5T + 8.47e11T^{2} \)
5 \( 1 + 4.14e8T + 2.98e17T^{2} \)
11 \( 1 - 7.00e12T + 1.08e26T^{2} \)
13 \( 1 - 1.39e13T + 7.05e27T^{2} \)
17 \( 1 - 2.13e15T + 5.77e30T^{2} \)
19 \( 1 - 9.59e15T + 9.30e31T^{2} \)
23 \( 1 + 5.44e16T + 1.10e34T^{2} \)
29 \( 1 + 1.99e18T + 3.63e36T^{2} \)
31 \( 1 - 4.95e18T + 1.92e37T^{2} \)
37 \( 1 - 3.59e19T + 1.60e39T^{2} \)
41 \( 1 + 1.40e20T + 2.08e40T^{2} \)
43 \( 1 + 2.16e19T + 6.86e40T^{2} \)
47 \( 1 + 6.68e19T + 6.34e41T^{2} \)
53 \( 1 - 6.44e21T + 1.27e43T^{2} \)
59 \( 1 - 1.61e22T + 1.86e44T^{2} \)
61 \( 1 - 1.05e22T + 4.29e44T^{2} \)
67 \( 1 - 3.25e22T + 4.48e45T^{2} \)
71 \( 1 - 1.13e23T + 1.91e46T^{2} \)
73 \( 1 + 4.97e22T + 3.82e46T^{2} \)
79 \( 1 - 7.02e23T + 2.75e47T^{2} \)
83 \( 1 - 6.80e23T + 9.48e47T^{2} \)
89 \( 1 + 3.93e23T + 5.42e48T^{2} \)
97 \( 1 - 1.28e25T + 4.66e49T^{2} \)
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

−11.57752611039507565721955403192, −9.716969974417592967867855087358, −8.572197392447718543556536042487, −7.53368145477126667493557705468, −6.14919446821576233524971445091, −5.17145933463604505771895616445, −3.84419094313783797254039467644, −3.44067939010728960453387228985, −2.28662642876879031217750703029, −0.68363839448233650155091006545, 0.68363839448233650155091006545, 2.28662642876879031217750703029, 3.44067939010728960453387228985, 3.84419094313783797254039467644, 5.17145933463604505771895616445, 6.14919446821576233524971445091, 7.53368145477126667493557705468, 8.572197392447718543556536042487, 9.716969974417592967867855087358, 11.57752611039507565721955403192

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