Properties

Label 2-8085-1.1-c1-0-100
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  − 2.41·2-s + 3-s + 3.82·4-s + 5-s − 2.41·6-s − 4.41·8-s + 9-s − 2.41·10-s − 11-s + 3.82·12-s + 5.65·13-s + 15-s + 2.99·16-s + 1.17·17-s − 2.41·18-s + 6.82·19-s + 3.82·20-s + 2.41·22-s − 4·23-s − 4.41·24-s + 25-s − 13.6·26-s + 27-s − 4.82·29-s − 2.41·30-s + 1.58·32-s − 33-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.91·4-s + 0.447·5-s − 0.985·6-s − 1.56·8-s + 0.333·9-s − 0.763·10-s − 0.301·11-s + 1.10·12-s + 1.56·13-s + 0.258·15-s + 0.749·16-s + 0.284·17-s − 0.569·18-s + 1.56·19-s + 0.856·20-s + 0.514·22-s − 0.834·23-s − 0.901·24-s + 0.200·25-s − 2.67·26-s + 0.192·27-s − 0.896·29-s − 0.440·30-s + 0.280·32-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: \(0\)
Selberg data: \((2,\ 8085,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445084191\)
\(L(\frac12)\) \(\approx\) \(1.445084191\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 2.34T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−7.925419787888793149562641091416, −7.50710586594100702950928824924, −6.70202027973124963999278136979, −6.00389369797315729981346851207, −5.29798926517950019313485618213, −4.00066331106684838849716205573, −3.22625515012726526816551293027, −2.34644151540959607782657199352, −1.52509007740516277347493077238, −0.808065135520453500186850518396, 0.808065135520453500186850518396, 1.52509007740516277347493077238, 2.34644151540959607782657199352, 3.22625515012726526816551293027, 4.00066331106684838849716205573, 5.29798926517950019313485618213, 6.00389369797315729981346851207, 6.70202027973124963999278136979, 7.50710586594100702950928824924, 7.925419787888793149562641091416

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