Properties

Label 2-8085-1.1-c1-0-106
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  + 1.30·2-s + 3-s − 0.288·4-s + 5-s + 1.30·6-s − 2.99·8-s + 9-s + 1.30·10-s + 11-s − 0.288·12-s + 0.562·13-s + 15-s − 3.34·16-s − 0.985·17-s + 1.30·18-s − 1.48·19-s − 0.288·20-s + 1.30·22-s − 5.24·23-s − 2.99·24-s + 25-s + 0.736·26-s + 27-s + 6.41·29-s + 1.30·30-s + 7.09·31-s + 1.61·32-s + ⋯
L(s)  = 1  + 0.925·2-s + 0.577·3-s − 0.144·4-s + 0.447·5-s + 0.534·6-s − 1.05·8-s + 0.333·9-s + 0.413·10-s + 0.301·11-s − 0.0832·12-s + 0.156·13-s + 0.258·15-s − 0.835·16-s − 0.239·17-s + 0.308·18-s − 0.341·19-s − 0.0644·20-s + 0.278·22-s − 1.09·23-s − 0.611·24-s + 0.200·25-s + 0.144·26-s + 0.192·27-s + 1.19·29-s + 0.238·30-s + 1.27·31-s + 0.286·32-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

\(L(1)\) \(\approx\) \(3.850728555\)
\(L(\frac12)\) \(\approx\) \(3.850728555\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.30T + 2T^{2} \)
13 \( 1 - 0.562T + 13T^{2} \)
17 \( 1 + 0.985T + 17T^{2} \)
19 \( 1 + 1.48T + 19T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 - 6.41T + 29T^{2} \)
31 \( 1 - 7.09T + 31T^{2} \)
37 \( 1 - 3.21T + 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 + 6.03T + 43T^{2} \)
47 \( 1 - 5.42T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 + 0.889T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 8.83T + 67T^{2} \)
71 \( 1 - 8.17T + 71T^{2} \)
73 \( 1 + 8.35T + 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 9.06T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.037293564234354080320674959579, −6.84561381766456558717590719894, −6.36375422811486411716369850750, −5.71851240814150937391244214942, −4.85360256227693538932168034409, −4.28666753561920669620204134024, −3.63321714900908417363792719113, −2.77658208951925427127074201408, −2.11629131858071353454246361179, −0.820390795340939821195549626865, 0.820390795340939821195549626865, 2.11629131858071353454246361179, 2.77658208951925427127074201408, 3.63321714900908417363792719113, 4.28666753561920669620204134024, 4.85360256227693538932168034409, 5.71851240814150937391244214942, 6.36375422811486411716369850750, 6.84561381766456558717590719894, 8.037293564234354080320674959579

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