Properties

Label 2-98-1.1-c3-0-7
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

\(L(2)\) \(\approx\) \(3.432688519\)
\(L(\frac12)\) \(\approx\) \(3.432688519\)
\(L(\frac{5}{2})\) 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

−13.59963797485483083534632429630, −12.65761964241543160645639262883, −11.60381728625889836671756696456, −10.00587691118305216349327883130, −8.894540399069061869708057750005, −7.79143107836201692503870437619, −6.94647134629169526302865347732, −4.59366733869488539456384880253, −3.56321171856643082977611247433, −2.26949176586125277843156415495, 2.26949176586125277843156415495, 3.56321171856643082977611247433, 4.59366733869488539456384880253, 6.94647134629169526302865347732, 7.79143107836201692503870437619, 8.894540399069061869708057750005, 10.00587691118305216349327883130, 11.60381728625889836671756696456, 12.65761964241543160645639262883, 13.59963797485483083534632429630

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