Properties

Label 2-7e2-1.1-c5-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.08·2-s + 10.0·3-s + 18.1·4-s + 79.8·5-s + 71.4·6-s − 97.9·8-s − 141.·9-s + 565.·10-s + 351.·11-s + 183.·12-s + 291.·13-s + 804.·15-s − 1.27e3·16-s + 370.·17-s − 1.00e3·18-s − 1.50e3·19-s + 1.45e3·20-s + 2.49e3·22-s − 425.·23-s − 987.·24-s + 3.24e3·25-s + 2.06e3·26-s − 3.87e3·27-s − 7.78e3·29-s + 5.70e3·30-s + 2.57e3·31-s − 5.89e3·32-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.646·3-s + 0.567·4-s + 1.42·5-s + 0.809·6-s − 0.541·8-s − 0.581·9-s + 1.78·10-s + 0.876·11-s + 0.367·12-s + 0.478·13-s + 0.923·15-s − 1.24·16-s + 0.310·17-s − 0.728·18-s − 0.956·19-s + 0.810·20-s + 1.09·22-s − 0.167·23-s − 0.350·24-s + 1.03·25-s + 0.599·26-s − 1.02·27-s − 1.71·29-s + 1.15·30-s + 0.481·31-s − 1.01·32-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.995019783\)
\(L(\frac12)\) \(\approx\) \(3.995019783\)
\(L(\frac{7}{2})\) 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 \( 1 - 7.08T + 32T^{2} \)
3 \( 1 - 10.0T + 243T^{2} \)
5 \( 1 - 79.8T + 3.12e3T^{2} \)
11 \( 1 - 351.T + 1.61e5T^{2} \)
13 \( 1 - 291.T + 3.71e5T^{2} \)
17 \( 1 - 370.T + 1.41e6T^{2} \)
19 \( 1 + 1.50e3T + 2.47e6T^{2} \)
23 \( 1 + 425.T + 6.43e6T^{2} \)
29 \( 1 + 7.78e3T + 2.05e7T^{2} \)
31 \( 1 - 2.57e3T + 2.86e7T^{2} \)
37 \( 1 - 739.T + 6.93e7T^{2} \)
41 \( 1 + 7.02e3T + 1.15e8T^{2} \)
43 \( 1 - 1.83e3T + 1.47e8T^{2} \)
47 \( 1 - 1.53e3T + 2.29e8T^{2} \)
53 \( 1 + 9.53e3T + 4.18e8T^{2} \)
59 \( 1 - 2.96e4T + 7.14e8T^{2} \)
61 \( 1 - 4.65e4T + 8.44e8T^{2} \)
67 \( 1 - 2.67e4T + 1.35e9T^{2} \)
71 \( 1 + 1.43e4T + 1.80e9T^{2} \)
73 \( 1 - 7.00e4T + 2.07e9T^{2} \)
79 \( 1 + 2.70e4T + 3.07e9T^{2} \)
83 \( 1 - 7.97e4T + 3.93e9T^{2} \)
89 \( 1 + 4.35e4T + 5.58e9T^{2} \)
97 \( 1 + 1.03e5T + 8.58e9T^{2} \)
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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

−14.31878321429291714800765399999, −13.64913374811210314858378195923, −12.79297713179705541576307453515, −11.35977846119517956365892102443, −9.648223823811190284022303785448, −8.679442646246045687763180557989, −6.42517346785260233871620965577, −5.48944517134452957758436702474, −3.71738055932699853720200953201, −2.18066979147607435516174234252, 2.18066979147607435516174234252, 3.71738055932699853720200953201, 5.48944517134452957758436702474, 6.42517346785260233871620965577, 8.679442646246045687763180557989, 9.648223823811190284022303785448, 11.35977846119517956365892102443, 12.79297713179705541576307453515, 13.64913374811210314858378195923, 14.31878321429291714800765399999

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