Properties

Label 2-84966-1.1-c1-0-73
Degree $2$
Conductor $84966$
Sign $-1$
Analytic cond. $678.456$
Root an. cond. $26.0472$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s + 5·11-s + 12-s − 3·15-s + 16-s + 18-s − 6·19-s − 3·20-s + 5·22-s − 2·23-s + 24-s + 4·25-s + 27-s − 9·29-s − 3·30-s + 3·31-s + 32-s + 5·33-s + 36-s + 6·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.50·11-s + 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.670·20-s + 1.06·22-s − 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 1.67·29-s − 0.547·30-s + 0.538·31-s + 0.176·32-s + 0.870·33-s + 1/6·36-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84966\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(678.456\)
Root analytic conductor: \(26.0472\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{84966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 84966,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
17 \( 1 \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 13 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.42594678042304, −13.64756949845604, −13.18953447417702, −12.70827538492506, −12.22089467064771, −11.71824415402083, −11.32409585372275, −11.03413047477633, −10.12676853278164, −9.762273311962332, −8.982477758759490, −8.590301073278896, −8.071593498213452, −7.595573334993660, −7.000172530057610, −6.514981480303264, −6.113795610740212, −5.201098522612876, −4.587616666718367, −4.012530871449792, −3.753001615884839, −3.339903985932742, −2.358633679396791, −1.858884970711911, −0.9869528034914277, 0, 0.9869528034914277, 1.858884970711911, 2.358633679396791, 3.339903985932742, 3.753001615884839, 4.012530871449792, 4.587616666718367, 5.201098522612876, 6.113795610740212, 6.514981480303264, 7.000172530057610, 7.595573334993660, 8.071593498213452, 8.590301073278896, 8.982477758759490, 9.762273311962332, 10.12676853278164, 11.03413047477633, 11.32409585372275, 11.71824415402083, 12.22089467064771, 12.70827538492506, 13.18953447417702, 13.64756949845604, 14.42594678042304

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