Properties

Label 2-8085-1.1-c1-0-241
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  + 1.73·2-s − 3-s + 0.999·4-s + 5-s − 1.73·6-s − 1.73·8-s + 9-s + 1.73·10-s − 11-s − 0.999·12-s + 1.46·13-s − 15-s − 5·16-s + 1.73·18-s + 1.46·19-s + 0.999·20-s − 1.73·22-s − 6.92·23-s + 1.73·24-s + 25-s + 2.53·26-s − 27-s + 3.46·29-s − 1.73·30-s − 2.92·31-s − 5.19·32-s + 33-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577·3-s + 0.499·4-s + 0.447·5-s − 0.707·6-s − 0.612·8-s + 0.333·9-s + 0.547·10-s − 0.301·11-s − 0.288·12-s + 0.406·13-s − 0.258·15-s − 1.25·16-s + 0.408·18-s + 0.335·19-s + 0.223·20-s − 0.369·22-s − 1.44·23-s + 0.353·24-s + 0.200·25-s + 0.497·26-s − 0.192·27-s + 0.643·29-s − 0.316·30-s − 0.525·31-s − 0.918·32-s + 0.174·33-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 \( 1 - 1.73T + 2T^{2} \)
13 \( 1 - 1.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 1.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 2.92T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−7.31434851072125990058655147365, −6.24981023799973078778279811393, −6.07962547639094169039755958500, −5.43592738551795314650104890813, −4.59204945088486997285361374960, −4.18498602597056118879939397501, −3.21719896217484920752980483116, −2.48716381127351422689302324811, −1.39498645130528337531736069637, 0, 1.39498645130528337531736069637, 2.48716381127351422689302324811, 3.21719896217484920752980483116, 4.18498602597056118879939397501, 4.59204945088486997285361374960, 5.43592738551795314650104890813, 6.07962547639094169039755958500, 6.24981023799973078778279811393, 7.31434851072125990058655147365

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