Properties

Label 2-8085-1.1-c1-0-14
Degree $2$
Conductor $8085$
Sign $1$
Analytic cond. $64.5590$
Root an. cond. $8.03486$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.811·2-s + 3-s − 1.34·4-s − 5-s − 0.811·6-s + 2.71·8-s + 9-s + 0.811·10-s + 11-s − 1.34·12-s − 5.06·13-s − 15-s + 0.479·16-s − 0.968·17-s − 0.811·18-s − 2.68·19-s + 1.34·20-s − 0.811·22-s − 5.21·23-s + 2.71·24-s + 25-s + 4.11·26-s + 27-s − 4.74·29-s + 0.811·30-s − 0.0517·31-s − 5.81·32-s + ⋯
L(s)  = 1  − 0.574·2-s + 0.577·3-s − 0.670·4-s − 0.447·5-s − 0.331·6-s + 0.958·8-s + 0.333·9-s + 0.256·10-s + 0.301·11-s − 0.387·12-s − 1.40·13-s − 0.258·15-s + 0.119·16-s − 0.234·17-s − 0.191·18-s − 0.615·19-s + 0.299·20-s − 0.173·22-s − 1.08·23-s + 0.553·24-s + 0.200·25-s + 0.807·26-s + 0.192·27-s − 0.881·29-s + 0.148·30-s − 0.00928·31-s − 1.02·32-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8085,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8155970124\)
\(L(\frac12)\) \(\approx\) \(0.8155970124\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.811T + 2T^{2} \)
13 \( 1 + 5.06T + 13T^{2} \)
17 \( 1 + 0.968T + 17T^{2} \)
19 \( 1 + 2.68T + 19T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 + 0.0517T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 - 3.96T + 41T^{2} \)
43 \( 1 + 5.06T + 43T^{2} \)
47 \( 1 - 4.05T + 47T^{2} \)
53 \( 1 - 1.97T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 1.29T + 67T^{2} \)
71 \( 1 + 8.59T + 71T^{2} \)
73 \( 1 - 14.2T + 73T^{2} \)
79 \( 1 + 8.67T + 79T^{2} \)
83 \( 1 - 4.77T + 83T^{2} \)
89 \( 1 - 4.09T + 89T^{2} \)
97 \( 1 + 0.0180T + 97T^{2} \)
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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

−7.86975615325059557979788947703, −7.43221250891918259910645446804, −6.71927443036135141980932195263, −5.68276471080845759238122750377, −4.84416382197467737847839931248, −4.21566886621655915969916677951, −3.65913645241451899043729728877, −2.52747143566848102095077280590, −1.75983981101741609632208248328, −0.46848598843894434694737368024, 0.46848598843894434694737368024, 1.75983981101741609632208248328, 2.52747143566848102095077280590, 3.65913645241451899043729728877, 4.21566886621655915969916677951, 4.84416382197467737847839931248, 5.68276471080845759238122750377, 6.71927443036135141980932195263, 7.43221250891918259910645446804, 7.86975615325059557979788947703

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