Properties

Label 2-8085-1.1-c1-0-127
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  + 0.111·2-s − 3-s − 1.98·4-s − 5-s − 0.111·6-s − 0.445·8-s + 9-s − 0.111·10-s − 11-s + 1.98·12-s − 1.18·13-s + 15-s + 3.92·16-s − 4.97·17-s + 0.111·18-s + 2.97·19-s + 1.98·20-s − 0.111·22-s − 4.29·23-s + 0.445·24-s + 25-s − 0.132·26-s − 27-s + 1.95·29-s + 0.111·30-s − 3.76·31-s + 1.32·32-s + ⋯
L(s)  = 1  + 0.0789·2-s − 0.577·3-s − 0.993·4-s − 0.447·5-s − 0.0456·6-s − 0.157·8-s + 0.333·9-s − 0.0353·10-s − 0.301·11-s + 0.573·12-s − 0.328·13-s + 0.258·15-s + 0.981·16-s − 1.20·17-s + 0.0263·18-s + 0.683·19-s + 0.444·20-s − 0.0238·22-s − 0.895·23-s + 0.0909·24-s + 0.200·25-s − 0.0259·26-s − 0.192·27-s + 0.363·29-s + 0.0203·30-s − 0.676·31-s + 0.234·32-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: \(1\)
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 \( 1 - 0.111T + 2T^{2} \)
13 \( 1 + 1.18T + 13T^{2} \)
17 \( 1 + 4.97T + 17T^{2} \)
19 \( 1 - 2.97T + 19T^{2} \)
23 \( 1 + 4.29T + 23T^{2} \)
29 \( 1 - 1.95T + 29T^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 - 8.28T + 37T^{2} \)
41 \( 1 - 9.93T + 41T^{2} \)
43 \( 1 - 5.93T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 1.74T + 53T^{2} \)
59 \( 1 - 9.95T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 - 4.36T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 7.00T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 1.62T + 97T^{2} \)
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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

−7.55781326784638012342161231707, −6.79278645031255007068549120182, −5.91181320387425188357410880093, −5.40578513156470531000919561396, −4.44814096649725797537492015144, −4.27171519650902749011373227857, −3.24830863691197766384622118522, −2.24111165161248985511343789420, −0.918081018576443815331498124078, 0, 0.918081018576443815331498124078, 2.24111165161248985511343789420, 3.24830863691197766384622118522, 4.27171519650902749011373227857, 4.44814096649725797537492015144, 5.40578513156470531000919561396, 5.91181320387425188357410880093, 6.79278645031255007068549120182, 7.55781326784638012342161231707

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