Properties

Label 2-98-1.1-c3-0-0
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 − 7.07·3-s + 4·4-s − 19.7·5-s + 14.1·6-s − 8·8-s + 23.0·9-s + 39.5·10-s − 14·11-s − 28.2·12-s + 50.9·13-s + 140·15-s + 16·16-s + 1.41·17-s − 46.0·18-s − 1.41·19-s − 79.1·20-s + 28·22-s + 140·23-s + 56.5·24-s + 267·25-s − 101.·26-s + 28.2·27-s − 286·29-s − 280·30-s − 93.3·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.36·3-s + 0.5·4-s − 1.77·5-s + 0.962·6-s − 0.353·8-s + 0.851·9-s + 1.25·10-s − 0.383·11-s − 0.680·12-s + 1.08·13-s + 2.40·15-s + 0.250·16-s + 0.0201·17-s − 0.602·18-s − 0.0170·19-s − 0.885·20-s + 0.271·22-s + 1.26·23-s + 0.481·24-s + 2.13·25-s − 0.768·26-s + 0.201·27-s − 1.83·29-s − 1.70·30-s − 0.540·31-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

\(L(2)\) \(\approx\) \(0.3748434021\)
\(L(\frac12)\) \(\approx\) \(0.3748434021\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.07T + 27T^{2} \)
5 \( 1 + 19.7T + 125T^{2} \)
11 \( 1 + 14T + 1.33e3T^{2} \)
13 \( 1 - 50.9T + 2.19e3T^{2} \)
17 \( 1 - 1.41T + 4.91e3T^{2} \)
19 \( 1 + 1.41T + 6.85e3T^{2} \)
23 \( 1 - 140T + 1.21e4T^{2} \)
29 \( 1 + 286T + 2.43e4T^{2} \)
31 \( 1 + 93.3T + 2.97e4T^{2} \)
37 \( 1 + 38T + 5.06e4T^{2} \)
41 \( 1 + 125.T + 6.89e4T^{2} \)
43 \( 1 + 34T + 7.95e4T^{2} \)
47 \( 1 - 523.T + 1.03e5T^{2} \)
53 \( 1 + 74T + 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 - 14.1T + 2.26e5T^{2} \)
67 \( 1 - 684T + 3.00e5T^{2} \)
71 \( 1 - 588T + 3.57e5T^{2} \)
73 \( 1 + 270.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 422.T + 5.71e5T^{2} \)
89 \( 1 - 618.T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−12.95967567290602917403137230869, −11.96728387137672545605707381280, −11.14181711493088547971662539773, −10.76558519542085117149843533826, −8.924645867876475589411123580714, −7.76394760220724446805700400063, −6.77108089612175459885657561003, −5.30684781566433497682552596931, −3.71050791864670579925600787107, −0.64076081636971140382714899099, 0.64076081636971140382714899099, 3.71050791864670579925600787107, 5.30684781566433497682552596931, 6.77108089612175459885657561003, 7.76394760220724446805700400063, 8.924645867876475589411123580714, 10.76558519542085117149843533826, 11.14181711493088547971662539773, 11.96728387137672545605707381280, 12.95967567290602917403137230869

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