Properties

Label 2-8085-1.1-c1-0-17
Degree $2$
Conductor $8085$
Sign $1$
Analytic cond. $64.5590$
Root an. cond. $8.03486$
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 − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s − 11-s + 12-s + 2·13-s + 15-s − 16-s − 2·17-s − 18-s − 4·19-s + 20-s + 22-s − 3·24-s + 25-s − 2·26-s − 27-s + 6·29-s − 30-s − 5·32-s + 33-s + 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s − 0.883·32-s + 0.174·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8085\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(64.5590\)
Root analytic conductor: \(8.03486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8085} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8085,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5930870866\)
\(L(\frac12)\) \(\approx\) \(0.5930870866\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.015635276180971767072104261450, −7.24122804370112636130661004061, −6.50262856054864000840404368005, −5.84265468518867538368638454369, −4.85320583518997291600325246890, −4.42803545304143239640744923294, −3.68523085800913719333483034712, −2.53363257502880775601913018184, −1.40311105147486487484331240797, −0.47637154236912241375530179655, 0.47637154236912241375530179655, 1.40311105147486487484331240797, 2.53363257502880775601913018184, 3.68523085800913719333483034712, 4.42803545304143239640744923294, 4.85320583518997291600325246890, 5.84265468518867538368638454369, 6.50262856054864000840404368005, 7.24122804370112636130661004061, 8.015635276180971767072104261450

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