Properties

Label 2-98-1.1-c3-0-4
Degree $2$
Conductor $98$
Sign $1$
Analytic cond. $5.78218$
Root an. cond. $2.40461$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 4·4-s + 12·5-s + 4·6-s + 8·8-s − 23·9-s + 24·10-s + 48·11-s + 8·12-s − 56·13-s + 24·15-s + 16·16-s + 114·17-s − 46·18-s − 2·19-s + 48·20-s + 96·22-s − 120·23-s + 16·24-s + 19·25-s − 112·26-s − 100·27-s − 54·29-s + 48·30-s − 236·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.384·3-s + 1/2·4-s + 1.07·5-s + 0.272·6-s + 0.353·8-s − 0.851·9-s + 0.758·10-s + 1.31·11-s + 0.192·12-s − 1.19·13-s + 0.413·15-s + 1/4·16-s + 1.62·17-s − 0.602·18-s − 0.0241·19-s + 0.536·20-s + 0.930·22-s − 1.08·23-s + 0.136·24-s + 0.151·25-s − 0.844·26-s − 0.712·27-s − 0.345·29-s + 0.292·30-s − 1.36·31-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.902371892\)
\(L(\frac12)\) \(\approx\) \(2.902371892\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
7 \( 1 \)
good3 \( 1 - 2 T + p^{3} T^{2} \)
5 \( 1 - 12 T + p^{3} T^{2} \)
11 \( 1 - 48 T + p^{3} T^{2} \)
13 \( 1 + 56 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 + 2 T + p^{3} T^{2} \)
23 \( 1 + 120 T + p^{3} T^{2} \)
29 \( 1 + 54 T + p^{3} T^{2} \)
31 \( 1 + 236 T + p^{3} T^{2} \)
37 \( 1 - 146 T + p^{3} T^{2} \)
41 \( 1 + 126 T + p^{3} T^{2} \)
43 \( 1 + 376 T + p^{3} T^{2} \)
47 \( 1 - 12 T + p^{3} T^{2} \)
53 \( 1 - 174 T + p^{3} T^{2} \)
59 \( 1 + 138 T + p^{3} T^{2} \)
61 \( 1 + 380 T + p^{3} T^{2} \)
67 \( 1 + 484 T + p^{3} T^{2} \)
71 \( 1 - 576 T + p^{3} T^{2} \)
73 \( 1 - 1150 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 + 378 T + p^{3} T^{2} \)
89 \( 1 - 390 T + p^{3} T^{2} \)
97 \( 1 - 1330 T + p^{3} T^{2} \)
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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

−13.73397897870439385301574992926, −12.41452635576454198119214831976, −11.61879940966370156439632035826, −10.06447456248675177451945348105, −9.252572613032144419263243224512, −7.71524436453679388439034948965, −6.25676915027756499817112600586, −5.30769600119045476453389118174, −3.51604532587599459406552309244, −1.98336991228660146622330664729, 1.98336991228660146622330664729, 3.51604532587599459406552309244, 5.30769600119045476453389118174, 6.25676915027756499817112600586, 7.71524436453679388439034948965, 9.252572613032144419263243224512, 10.06447456248675177451945348105, 11.61879940966370156439632035826, 12.41452635576454198119214831976, 13.73397897870439385301574992926

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