Properties

Label 2-98-1.1-c9-0-23
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $50.4735$
Root an. cond. $7.10447$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 270.·3-s + 256·4-s + 1.36e3·5-s + 4.33e3·6-s + 4.09e3·8-s + 5.36e4·9-s + 2.19e4·10-s − 1.09e4·11-s + 6.93e4·12-s − 7.96e3·13-s + 3.70e5·15-s + 6.55e4·16-s − 3.24e5·17-s + 8.58e5·18-s − 2.31e5·19-s + 3.50e5·20-s − 1.74e5·22-s + 2.10e6·23-s + 1.10e6·24-s − 7.74e4·25-s − 1.27e5·26-s + 9.20e6·27-s − 5.46e6·29-s + 5.93e6·30-s − 1.97e6·31-s + 1.04e6·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.93·3-s + 0.5·4-s + 0.979·5-s + 1.36·6-s + 0.353·8-s + 2.72·9-s + 0.692·10-s − 0.224·11-s + 0.965·12-s − 0.0773·13-s + 1.89·15-s + 0.250·16-s − 0.940·17-s + 1.92·18-s − 0.408·19-s + 0.489·20-s − 0.159·22-s + 1.56·23-s + 0.682·24-s − 0.0396·25-s − 0.0547·26-s + 3.33·27-s − 1.43·29-s + 1.33·30-s − 0.384·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(50.4735\)
Root analytic conductor: \(7.10447\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(7.883762343\)
\(L(\frac12)\) \(\approx\) \(7.883762343\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 16T \)
7 \( 1 \)
good3 \( 1 - 270.T + 1.96e4T^{2} \)
5 \( 1 - 1.36e3T + 1.95e6T^{2} \)
11 \( 1 + 1.09e4T + 2.35e9T^{2} \)
13 \( 1 + 7.96e3T + 1.06e10T^{2} \)
17 \( 1 + 3.24e5T + 1.18e11T^{2} \)
19 \( 1 + 2.31e5T + 3.22e11T^{2} \)
23 \( 1 - 2.10e6T + 1.80e12T^{2} \)
29 \( 1 + 5.46e6T + 1.45e13T^{2} \)
31 \( 1 + 1.97e6T + 2.64e13T^{2} \)
37 \( 1 - 4.76e6T + 1.29e14T^{2} \)
41 \( 1 - 5.57e6T + 3.27e14T^{2} \)
43 \( 1 + 3.11e7T + 5.02e14T^{2} \)
47 \( 1 - 4.31e6T + 1.11e15T^{2} \)
53 \( 1 + 9.02e7T + 3.29e15T^{2} \)
59 \( 1 + 7.29e7T + 8.66e15T^{2} \)
61 \( 1 + 7.43e6T + 1.16e16T^{2} \)
67 \( 1 - 1.05e8T + 2.72e16T^{2} \)
71 \( 1 - 1.10e8T + 4.58e16T^{2} \)
73 \( 1 - 2.04e8T + 5.88e16T^{2} \)
79 \( 1 - 8.87e7T + 1.19e17T^{2} \)
83 \( 1 - 2.41e8T + 1.86e17T^{2} \)
89 \( 1 - 5.48e8T + 3.50e17T^{2} \)
97 \( 1 + 1.16e9T + 7.60e17T^{2} \)
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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

−12.87562364441926446746276109928, −10.90854856062389466044211373559, −9.668350605724094584073888359767, −8.932055737598557278999216396641, −7.68667682913807506803841566493, −6.57382117571000237553778493140, −4.88139256413102479736549949455, −3.57048927424647125138485048197, −2.47651340214972685233681464009, −1.67434932661066449156776031991, 1.67434932661066449156776031991, 2.47651340214972685233681464009, 3.57048927424647125138485048197, 4.88139256413102479736549949455, 6.57382117571000237553778493140, 7.68667682913807506803841566493, 8.932055737598557278999216396641, 9.668350605724094584073888359767, 10.90854856062389466044211373559, 12.87562364441926446746276109928

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