Properties

Label 2-98-1.1-c3-0-2
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 9.38·3-s + 4·4-s + 9.38·5-s − 18.7·6-s + 8·8-s + 61·9-s + 18.7·10-s + 20·11-s − 37.5·12-s + 65.6·13-s − 88·15-s + 16·16-s + 56.2·17-s + 122·18-s + 9.38·19-s + 37.5·20-s + 40·22-s + 48·23-s − 75.0·24-s − 37·25-s + 131.·26-s − 318.·27-s − 166·29-s − 176·30-s − 206.·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.839·5-s − 1.27·6-s + 0.353·8-s + 2.25·9-s + 0.593·10-s + 0.548·11-s − 0.902·12-s + 1.40·13-s − 1.51·15-s + 0.250·16-s + 0.803·17-s + 1.59·18-s + 0.113·19-s + 0.419·20-s + 0.387·22-s + 0.435·23-s − 0.638·24-s − 0.295·25-s + 0.990·26-s − 2.27·27-s − 1.06·29-s − 1.07·30-s − 1.19·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98\)    =    \(2 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(5.78218\)
Root analytic conductor: \(2.40461\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{98} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.678121860\)
\(L(\frac12)\) \(\approx\) \(1.678121860\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
7 \( 1 \)
good3 \( 1 + 9.38T + 27T^{2} \)
5 \( 1 - 9.38T + 125T^{2} \)
11 \( 1 - 20T + 1.33e3T^{2} \)
13 \( 1 - 65.6T + 2.19e3T^{2} \)
17 \( 1 - 56.2T + 4.91e3T^{2} \)
19 \( 1 - 9.38T + 6.85e3T^{2} \)
23 \( 1 - 48T + 1.21e4T^{2} \)
29 \( 1 + 166T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + 78T + 5.06e4T^{2} \)
41 \( 1 - 393.T + 6.89e4T^{2} \)
43 \( 1 - 436T + 7.95e4T^{2} \)
47 \( 1 - 206.T + 1.03e5T^{2} \)
53 \( 1 - 62T + 1.48e5T^{2} \)
59 \( 1 + 666.T + 2.05e5T^{2} \)
61 \( 1 - 272.T + 2.26e5T^{2} \)
67 \( 1 - 580T + 3.00e5T^{2} \)
71 \( 1 + 544T + 3.57e5T^{2} \)
73 \( 1 + 600.T + 3.89e5T^{2} \)
79 \( 1 + 680T + 4.93e5T^{2} \)
83 \( 1 - 196.T + 5.71e5T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + 656.T + 9.12e5T^{2} \)
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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

−13.18031487263087406742530126025, −12.39851490830612553633858556115, −11.29996879555470691184140180085, −10.69905435151612401309135613329, −9.418810470744951450183158449036, −7.22634835454322682372038762076, −5.96000826408606816820597607154, −5.62322633178832773910526720905, −4.03583657467447267655429694013, −1.33862522665358230379204874497, 1.33862522665358230379204874497, 4.03583657467447267655429694013, 5.62322633178832773910526720905, 5.96000826408606816820597607154, 7.22634835454322682372038762076, 9.418810470744951450183158449036, 10.69905435151612401309135613329, 11.29996879555470691184140180085, 12.39851490830612553633858556115, 13.18031487263087406742530126025

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