Properties

Label 2-98-1.1-c9-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s − 123.·3-s + 256·4-s + 1.27e3·5-s − 1.97e3·6-s + 4.09e3·8-s − 4.41e3·9-s + 2.04e4·10-s − 2.76e4·11-s − 3.16e4·12-s − 8.33e4·13-s − 1.57e5·15-s + 6.55e4·16-s + 5.54e5·17-s − 7.07e4·18-s − 3.43e5·19-s + 3.26e5·20-s − 4.42e5·22-s − 5.35e5·23-s − 5.06e5·24-s − 3.23e5·25-s − 1.33e6·26-s + 2.97e6·27-s − 2.59e6·29-s − 2.52e6·30-s + 6.09e6·31-s + 1.04e6·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.880·3-s + 0.5·4-s + 0.913·5-s − 0.622·6-s + 0.353·8-s − 0.224·9-s + 0.645·10-s − 0.569·11-s − 0.440·12-s − 0.809·13-s − 0.804·15-s + 0.250·16-s + 1.60·17-s − 0.158·18-s − 0.604·19-s + 0.456·20-s − 0.402·22-s − 0.399·23-s − 0.311·24-s − 0.165·25-s − 0.572·26-s + 1.07·27-s − 0.680·29-s − 0.568·30-s + 1.18·31-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: \(1\)
Selberg data: \((2,\ 98,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 + 123.T + 1.96e4T^{2} \)
5 \( 1 - 1.27e3T + 1.95e6T^{2} \)
11 \( 1 + 2.76e4T + 2.35e9T^{2} \)
13 \( 1 + 8.33e4T + 1.06e10T^{2} \)
17 \( 1 - 5.54e5T + 1.18e11T^{2} \)
19 \( 1 + 3.43e5T + 3.22e11T^{2} \)
23 \( 1 + 5.35e5T + 1.80e12T^{2} \)
29 \( 1 + 2.59e6T + 1.45e13T^{2} \)
31 \( 1 - 6.09e6T + 2.64e13T^{2} \)
37 \( 1 + 1.78e7T + 1.29e14T^{2} \)
41 \( 1 + 3.05e7T + 3.27e14T^{2} \)
43 \( 1 + 8.75e6T + 5.02e14T^{2} \)
47 \( 1 + 2.72e7T + 1.11e15T^{2} \)
53 \( 1 + 8.34e7T + 3.29e15T^{2} \)
59 \( 1 + 8.15e7T + 8.66e15T^{2} \)
61 \( 1 + 4.98e7T + 1.16e16T^{2} \)
67 \( 1 + 1.85e8T + 2.72e16T^{2} \)
71 \( 1 - 3.70e8T + 4.58e16T^{2} \)
73 \( 1 - 2.46e8T + 5.88e16T^{2} \)
79 \( 1 + 5.19e8T + 1.19e17T^{2} \)
83 \( 1 - 2.98e8T + 1.86e17T^{2} \)
89 \( 1 + 2.97e8T + 3.50e17T^{2} \)
97 \( 1 + 7.80e8T + 7.60e17T^{2} \)
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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

−11.82236854078292406429467246080, −10.55728209305113004763534349412, −9.815060782048062711010606091129, −8.048047927829842809235348544844, −6.60159460989186824240392611696, −5.62952361146217294659237088467, −4.94922883066078126700334948675, −3.11480563363481081390916390525, −1.72991262628657713484677009694, 0, 1.72991262628657713484677009694, 3.11480563363481081390916390525, 4.94922883066078126700334948675, 5.62952361146217294659237088467, 6.60159460989186824240392611696, 8.048047927829842809235348544844, 9.815060782048062711010606091129, 10.55728209305113004763534349412, 11.82236854078292406429467246080

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