Properties

Label 2-7e2-1.1-c5-0-8
Degree $2$
Conductor $49$
Sign $-1$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.81·2-s + 23.5·3-s + 29.0·4-s − 74.2·5-s − 183.·6-s + 22.8·8-s + 310.·9-s + 580.·10-s − 424.·11-s + 683.·12-s − 252.·13-s − 1.74e3·15-s − 1.10e3·16-s − 1.10e3·17-s − 2.42e3·18-s − 6.47·19-s − 2.15e3·20-s + 3.31e3·22-s − 3.61e3·23-s + 537.·24-s + 2.39e3·25-s + 1.97e3·26-s + 1.57e3·27-s − 5.00e3·29-s + 1.36e4·30-s + 2.82e3·31-s + 7.93e3·32-s + ⋯
L(s)  = 1  − 1.38·2-s + 1.50·3-s + 0.908·4-s − 1.32·5-s − 2.08·6-s + 0.126·8-s + 1.27·9-s + 1.83·10-s − 1.05·11-s + 1.37·12-s − 0.413·13-s − 2.00·15-s − 1.08·16-s − 0.926·17-s − 1.76·18-s − 0.00411·19-s − 1.20·20-s + 1.46·22-s − 1.42·23-s + 0.190·24-s + 0.765·25-s + 0.571·26-s + 0.416·27-s − 1.10·29-s + 2.76·30-s + 0.527·31-s + 1.36·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-1$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ -1)\)

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)$
bad7 \( 1 \)
good2 \( 1 + 7.81T + 32T^{2} \)
3 \( 1 - 23.5T + 243T^{2} \)
5 \( 1 + 74.2T + 3.12e3T^{2} \)
11 \( 1 + 424.T + 1.61e5T^{2} \)
13 \( 1 + 252.T + 3.71e5T^{2} \)
17 \( 1 + 1.10e3T + 1.41e6T^{2} \)
19 \( 1 + 6.47T + 2.47e6T^{2} \)
23 \( 1 + 3.61e3T + 6.43e6T^{2} \)
29 \( 1 + 5.00e3T + 2.05e7T^{2} \)
31 \( 1 - 2.82e3T + 2.86e7T^{2} \)
37 \( 1 + 2.04e3T + 6.93e7T^{2} \)
41 \( 1 - 9.39e3T + 1.15e8T^{2} \)
43 \( 1 - 1.03e4T + 1.47e8T^{2} \)
47 \( 1 - 1.70e4T + 2.29e8T^{2} \)
53 \( 1 + 3.95e4T + 4.18e8T^{2} \)
59 \( 1 - 3.39e4T + 7.14e8T^{2} \)
61 \( 1 - 2.82e4T + 8.44e8T^{2} \)
67 \( 1 - 5.61e4T + 1.35e9T^{2} \)
71 \( 1 + 1.55e4T + 1.80e9T^{2} \)
73 \( 1 + 7.82e4T + 2.07e9T^{2} \)
79 \( 1 + 4.53e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e3T + 3.93e9T^{2} \)
89 \( 1 + 6.88e4T + 5.58e9T^{2} \)
97 \( 1 - 1.08e5T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.22958765460371628309945597813, −12.93888072460936229013246458965, −11.31220542899539460003196452054, −10.02812523454651323689647803562, −8.863474104461205553391392585154, −7.982685497822485041401056898698, −7.42870472198819141502041605806, −4.11671663151358339426894880399, −2.35933728263432701841238890333, 0, 2.35933728263432701841238890333, 4.11671663151358339426894880399, 7.42870472198819141502041605806, 7.982685497822485041401056898698, 8.863474104461205553391392585154, 10.02812523454651323689647803562, 11.31220542899539460003196452054, 12.93888072460936229013246458965, 14.22958765460371628309945597813

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