Properties

Label 2-8085-1.1-c1-0-214
Degree $2$
Conductor $8085$
Sign $-1$
Analytic cond. $64.5590$
Root an. cond. $8.03486$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s + 9-s + 2·10-s + 11-s − 2·12-s + 6·13-s + 15-s − 4·16-s + 7·17-s − 2·18-s + 5·19-s − 2·20-s − 2·22-s − 23-s + 25-s − 12·26-s − 27-s − 5·29-s − 2·30-s + 8·31-s + 8·32-s − 33-s − 14·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 1.66·13-s + 0.258·15-s − 16-s + 1.69·17-s − 0.471·18-s + 1.14·19-s − 0.447·20-s − 0.426·22-s − 0.208·23-s + 1/5·25-s − 2.35·26-s − 0.192·27-s − 0.928·29-s − 0.365·30-s + 1.43·31-s + 1.41·32-s − 0.174·33-s − 2.40·34-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: yes
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 \( 1 + p T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−7.73392688783567043999716677888, −6.89967820238640478935483111479, −6.36636118354222571650852423661, −5.50265839507652826423150297551, −4.78489530768419459469792706632, −3.69036506995419110474959413590, −3.17440742850973981254141522910, −1.48093151991038961683849338922, −1.23615984375973552403829036226, 0, 1.23615984375973552403829036226, 1.48093151991038961683849338922, 3.17440742850973981254141522910, 3.69036506995419110474959413590, 4.78489530768419459469792706632, 5.50265839507652826423150297551, 6.36636118354222571650852423661, 6.89967820238640478935483111479, 7.73392688783567043999716677888

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