Properties

Label 2-98-1.1-c9-0-3
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $50.4735$
Root an. cond. $7.10447$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s + 6·3-s + 256·4-s − 560·5-s − 96·6-s − 4.09e3·8-s − 1.96e4·9-s + 8.96e3·10-s − 5.41e4·11-s + 1.53e3·12-s + 1.13e5·13-s − 3.36e3·15-s + 6.55e4·16-s − 6.26e3·17-s + 3.14e5·18-s − 2.57e5·19-s − 1.43e5·20-s + 8.66e5·22-s − 2.66e5·23-s − 2.45e4·24-s − 1.63e6·25-s − 1.81e6·26-s − 2.35e5·27-s + 1.57e6·29-s + 5.37e4·30-s + 4.63e6·31-s − 1.04e6·32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0427·3-s + 1/2·4-s − 0.400·5-s − 0.0302·6-s − 0.353·8-s − 0.998·9-s + 0.283·10-s − 1.11·11-s + 0.0213·12-s + 1.09·13-s − 0.0171·15-s + 1/4·16-s − 0.0181·17-s + 0.705·18-s − 0.452·19-s − 0.200·20-s + 0.788·22-s − 0.198·23-s − 0.0151·24-s − 0.839·25-s − 0.777·26-s − 0.0854·27-s + 0.413·29-s + 0.0121·30-s + 0.901·31-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: yes
Arithmetic: yes
Character: $\chi_{98} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.8623924359\)
\(L(\frac12)\) \(\approx\) \(0.8623924359\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{4} T \)
7 \( 1 \)
good3 \( 1 - 2 p T + p^{9} T^{2} \)
5 \( 1 + 112 p T + p^{9} T^{2} \)
11 \( 1 + 54152 T + p^{9} T^{2} \)
13 \( 1 - 113172 T + p^{9} T^{2} \)
17 \( 1 + 6262 T + p^{9} T^{2} \)
19 \( 1 + 257078 T + p^{9} T^{2} \)
23 \( 1 + 266000 T + p^{9} T^{2} \)
29 \( 1 - 1574714 T + p^{9} T^{2} \)
31 \( 1 - 4637484 T + p^{9} T^{2} \)
37 \( 1 + 11946238 T + p^{9} T^{2} \)
41 \( 1 + 21909126 T + p^{9} T^{2} \)
43 \( 1 - 27520592 T + p^{9} T^{2} \)
47 \( 1 + 52927836 T + p^{9} T^{2} \)
53 \( 1 - 16221222 T + p^{9} T^{2} \)
59 \( 1 - 140509618 T + p^{9} T^{2} \)
61 \( 1 - 202963560 T + p^{9} T^{2} \)
67 \( 1 - 153734572 T + p^{9} T^{2} \)
71 \( 1 - 3938816 p T + p^{9} T^{2} \)
73 \( 1 - 404022830 T + p^{9} T^{2} \)
79 \( 1 + 130689816 T + p^{9} T^{2} \)
83 \( 1 + 420134014 T + p^{9} T^{2} \)
89 \( 1 - 469542390 T + p^{9} T^{2} \)
97 \( 1 - 872501690 T + p^{9} T^{2} \)
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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

−11.80289162470236328392684818027, −10.99573939270742659019862743996, −9.997008587010808902567677584213, −8.534185214240394812050401439458, −8.075504828464256155301875018104, −6.57524699634725858050024964994, −5.36276267278573117445148711528, −3.55888442102021507792682476254, −2.25451081391539111389523507839, −0.55391693837960335332937966589, 0.55391693837960335332937966589, 2.25451081391539111389523507839, 3.55888442102021507792682476254, 5.36276267278573117445148711528, 6.57524699634725858050024964994, 8.075504828464256155301875018104, 8.534185214240394812050401439458, 9.997008587010808902567677584213, 10.99573939270742659019862743996, 11.80289162470236328392684818027

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