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Properties

Label 2-98315-1.1-c1-0-2
Degree $2$
Conductor $98315$
Sign $1$
Analytic cond. $785.049$
Root an. cond. $28.0187$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s − 7-s − 2·9-s + 5·11-s − 2·12-s + 13-s + 15-s + 4·16-s + 3·17-s + 2·19-s − 2·20-s − 21-s + 4·23-s + 25-s − 5·27-s + 2·28-s − 29-s − 2·31-s + 5·33-s − 35-s + 4·36-s + 8·37-s + 39-s + 8·41-s − 8·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.962·27-s + 0.377·28-s − 0.185·29-s − 0.359·31-s + 0.870·33-s − 0.169·35-s + 2/3·36-s + 1.31·37-s + 0.160·39-s + 1.24·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98315\)    =    \(5 \cdot 7 \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(785.049\)
Root analytic conductor: \(28.0187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 98315,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.551249561\)
\(L(\frac12)\) \(\approx\) \(3.551249561\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 + T \)
53 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.87871627481664, −13.27307782932540, −13.05655542435047, −12.27704017771831, −12.01057377372747, −11.21553601836228, −10.94417621660741, −10.04184031030150, −9.596164560925102, −9.305714892019438, −8.974745944560242, −8.341773324983754, −7.968812509418662, −7.267975074354582, −6.612012704604987, −6.140164497692936, −5.417353846047639, −5.253636421247211, −4.242769024679762, −3.812214933730559, −3.395987687015730, −2.735755304097725, −2.011919710022394, −1.035311671443184, −0.7207609602107455, 0.7207609602107455, 1.035311671443184, 2.011919710022394, 2.735755304097725, 3.395987687015730, 3.812214933730559, 4.242769024679762, 5.253636421247211, 5.417353846047639, 6.140164497692936, 6.612012704604987, 7.267975074354582, 7.968812509418662, 8.341773324983754, 8.974745944560242, 9.305714892019438, 9.596164560925102, 10.04184031030150, 10.94417621660741, 11.21553601836228, 12.01057377372747, 12.27704017771831, 13.05655542435047, 13.27307782932540, 13.87871627481664

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