Properties

Label 2-8619-1.1-c1-0-120
Degree $2$
Conductor $8619$
Sign $1$
Analytic cond. $68.8230$
Root an. cond. $8.29596$
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  + 2·2-s − 3-s + 2·4-s − 2·5-s − 2·6-s + 4·7-s + 9-s − 4·10-s − 3·11-s − 2·12-s + 8·14-s + 2·15-s − 4·16-s + 17-s + 2·18-s − 7·19-s − 4·20-s − 4·21-s − 6·22-s − 25-s − 27-s + 8·28-s + 9·29-s + 4·30-s + 8·31-s − 8·32-s + 3·33-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s + 1.51·7-s + 1/3·9-s − 1.26·10-s − 0.904·11-s − 0.577·12-s + 2.13·14-s + 0.516·15-s − 16-s + 0.242·17-s + 0.471·18-s − 1.60·19-s − 0.894·20-s − 0.872·21-s − 1.27·22-s − 1/5·25-s − 0.192·27-s + 1.51·28-s + 1.67·29-s + 0.730·30-s + 1.43·31-s − 1.41·32-s + 0.522·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8619\)    =    \(3 \cdot 13^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(68.8230\)
Root analytic conductor: \(8.29596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8619,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.940650029\)
\(L(\frac12)\) \(\approx\) \(2.940650029\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−7.70224547312158870414838139451, −6.88644089951924194845284458992, −6.20963882703000360047142037294, −5.46413829867624494532195544092, −4.79703434369813196275546290927, −4.46486231340171915326858035439, −3.86265505185246303124129992521, −2.77462754106791638079684385007, −2.05406424872503173743387964745, −0.67993303887674977063741634232, 0.67993303887674977063741634232, 2.05406424872503173743387964745, 2.77462754106791638079684385007, 3.86265505185246303124129992521, 4.46486231340171915326858035439, 4.79703434369813196275546290927, 5.46413829867624494532195544092, 6.20963882703000360047142037294, 6.88644089951924194845284458992, 7.70224547312158870414838139451

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