Properties

Label 2-8085-1.1-c1-0-140
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.193·2-s − 3-s − 1.96·4-s − 5-s + 0.193·6-s + 0.768·8-s + 9-s + 0.193·10-s + 11-s + 1.96·12-s − 2.96·13-s + 15-s + 3.77·16-s + 4.57·17-s − 0.193·18-s + 4.31·19-s + 1.96·20-s − 0.193·22-s − 6.70·23-s − 0.768·24-s + 25-s + 0.574·26-s − 27-s − 3.61·29-s − 0.193·30-s − 9.92·31-s − 2.26·32-s + ⋯
L(s)  = 1  − 0.137·2-s − 0.577·3-s − 0.981·4-s − 0.447·5-s + 0.0791·6-s + 0.271·8-s + 0.333·9-s + 0.0613·10-s + 0.301·11-s + 0.566·12-s − 0.821·13-s + 0.258·15-s + 0.943·16-s + 1.10·17-s − 0.0457·18-s + 0.989·19-s + 0.438·20-s − 0.0413·22-s − 1.39·23-s − 0.156·24-s + 0.200·25-s + 0.112·26-s − 0.192·27-s − 0.670·29-s − 0.0354·30-s − 1.78·31-s − 0.401·32-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: $\chi_{8085} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8085,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 0.193T + 2T^{2} \)
13 \( 1 + 2.96T + 13T^{2} \)
17 \( 1 - 4.57T + 17T^{2} \)
19 \( 1 - 4.31T + 19T^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 + 9.92T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
43 \( 1 + 9.27T + 43T^{2} \)
47 \( 1 - 9.92T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 5.92T + 67T^{2} \)
71 \( 1 - 9.92T + 71T^{2} \)
73 \( 1 - 7.73T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 2.77T + 89T^{2} \)
97 \( 1 + 0.0752T + 97T^{2} \)
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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

−7.60935619787766945416664815359, −6.95220989958096284852928732591, −5.82798493486632127552674572292, −5.40005279022620948134281612004, −4.76384821168777503191246505266, −3.81463237168073284946123957412, −3.49355776959591115114122960015, −2.05674947715310654660218251995, −0.960093504784328200852568351606, 0, 0.960093504784328200852568351606, 2.05674947715310654660218251995, 3.49355776959591115114122960015, 3.81463237168073284946123957412, 4.76384821168777503191246505266, 5.40005279022620948134281612004, 5.82798493486632127552674572292, 6.95220989958096284852928732591, 7.60935619787766945416664815359

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