Properties

Label 2-85-1.1-c1-0-3
Degree $2$
Conductor $85$
Sign $1$
Analytic cond. $0.678728$
Root an. cond. $0.823849$
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-s + 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s + 2·11-s − 2·12-s + 2·13-s − 2·14-s − 2·15-s − 16-s + 17-s + 18-s + 20-s − 4·21-s + 2·22-s + 6·23-s − 6·24-s + 25-s + 2·26-s − 4·27-s + 2·28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.577·12-s + 0.554·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.223·20-s − 0.872·21-s + 0.426·22-s + 1.25·23-s − 1.22·24-s + 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85\)    =    \(5 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.678728\)
Root analytic conductor: \(0.823849\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.397692218\)
\(L(\frac12)\) \(\approx\) \(1.397692218\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
17 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.32499099465889905092766203697, −13.23869509800644511047101744766, −12.61695801969404065473449905589, −11.16926979542542511914194656047, −9.342260097121774077335621975199, −8.918128413786971033454816799413, −7.46613087643891033346359456413, −5.83398257972728633601666952176, −4.03297779911247219461657332614, −3.13757525842324917581154074415, 3.13757525842324917581154074415, 4.03297779911247219461657332614, 5.83398257972728633601666952176, 7.46613087643891033346359456413, 8.918128413786971033454816799413, 9.342260097121774077335621975199, 11.16926979542542511914194656047, 12.61695801969404065473449905589, 13.23869509800644511047101744766, 14.32499099465889905092766203697

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