Properties

Label 2-7e2-1.1-c25-0-10
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  + 300.·2-s − 1.52e6·3-s − 3.34e7·4-s − 9.44e8·5-s − 4.59e8·6-s − 2.01e10·8-s + 1.49e12·9-s − 2.83e11·10-s − 2.47e12·11-s + 5.11e13·12-s + 6.66e13·13-s + 1.44e15·15-s + 1.11e15·16-s + 2.96e15·17-s + 4.48e14·18-s − 5.82e15·19-s + 3.16e16·20-s − 7.42e14·22-s − 1.71e17·23-s + 3.07e16·24-s + 5.93e17·25-s + 2.00e16·26-s − 9.87e17·27-s + 3.51e18·29-s + 4.33e17·30-s − 5.01e18·31-s + 1.01e18·32-s + ⋯
L(s)  = 1  + 0.0518·2-s − 1.66·3-s − 0.997·4-s − 1.72·5-s − 0.0861·6-s − 0.103·8-s + 1.76·9-s − 0.0897·10-s − 0.237·11-s + 1.65·12-s + 0.793·13-s + 2.87·15-s + 0.991·16-s + 1.23·17-s + 0.0913·18-s − 0.603·19-s + 1.72·20-s − 0.0123·22-s − 1.62·23-s + 0.172·24-s + 1.99·25-s + 0.0411·26-s − 1.26·27-s + 1.84·29-s + 0.149·30-s − 1.14·31-s + 0.154·32-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

\(L(13)\) \(\approx\) \(0.3077604381\)
\(L(\frac12)\) \(\approx\) \(0.3077604381\)
\(L(\frac{27}{2})\) 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 \( 1 - 300.T + 3.35e7T^{2} \)
3 \( 1 + 1.52e6T + 8.47e11T^{2} \)
5 \( 1 + 9.44e8T + 2.98e17T^{2} \)
11 \( 1 + 2.47e12T + 1.08e26T^{2} \)
13 \( 1 - 6.66e13T + 7.05e27T^{2} \)
17 \( 1 - 2.96e15T + 5.77e30T^{2} \)
19 \( 1 + 5.82e15T + 9.30e31T^{2} \)
23 \( 1 + 1.71e17T + 1.10e34T^{2} \)
29 \( 1 - 3.51e18T + 3.63e36T^{2} \)
31 \( 1 + 5.01e18T + 1.92e37T^{2} \)
37 \( 1 + 1.07e19T + 1.60e39T^{2} \)
41 \( 1 + 2.09e20T + 2.08e40T^{2} \)
43 \( 1 - 2.88e20T + 6.86e40T^{2} \)
47 \( 1 - 6.92e20T + 6.34e41T^{2} \)
53 \( 1 - 1.61e21T + 1.27e43T^{2} \)
59 \( 1 + 1.36e22T + 1.86e44T^{2} \)
61 \( 1 + 2.85e22T + 4.29e44T^{2} \)
67 \( 1 - 5.59e22T + 4.48e45T^{2} \)
71 \( 1 - 2.07e23T + 1.91e46T^{2} \)
73 \( 1 - 3.73e22T + 3.82e46T^{2} \)
79 \( 1 - 4.44e23T + 2.75e47T^{2} \)
83 \( 1 - 1.41e22T + 9.48e47T^{2} \)
89 \( 1 + 2.95e24T + 5.42e48T^{2} \)
97 \( 1 + 1.91e23T + 4.66e49T^{2} \)
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

−11.00425912631038171132595640886, −10.17529116240331577309142582796, −8.489797206389792864876736805144, −7.61620485298476233134898431709, −6.24769562696473746603058587133, −5.19610847800074087596596826729, −4.27338780275735890529983478938, −3.56160747770264846211163639128, −1.09897542637285295566838796580, −0.32324714189595986379688132639, 0.32324714189595986379688132639, 1.09897542637285295566838796580, 3.56160747770264846211163639128, 4.27338780275735890529983478938, 5.19610847800074087596596826729, 6.24769562696473746603058587133, 7.61620485298476233134898431709, 8.489797206389792864876736805144, 10.17529116240331577309142582796, 11.00425912631038171132595640886

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