Properties

Label 2-8085-1.1-c1-0-128
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  + 0.475·2-s − 3-s − 1.77·4-s − 5-s − 0.475·6-s − 1.79·8-s + 9-s − 0.475·10-s − 11-s + 1.77·12-s − 6.31·13-s + 15-s + 2.69·16-s + 1.92·17-s + 0.475·18-s − 3.92·19-s + 1.77·20-s − 0.475·22-s + 4.74·23-s + 1.79·24-s + 25-s − 3.00·26-s − 27-s − 4.08·29-s + 0.475·30-s + 7.20·31-s + 4.87·32-s + ⋯
L(s)  = 1  + 0.336·2-s − 0.577·3-s − 0.886·4-s − 0.447·5-s − 0.194·6-s − 0.634·8-s + 0.333·9-s − 0.150·10-s − 0.301·11-s + 0.511·12-s − 1.75·13-s + 0.258·15-s + 0.673·16-s + 0.466·17-s + 0.112·18-s − 0.899·19-s + 0.396·20-s − 0.101·22-s + 0.988·23-s + 0.366·24-s + 0.200·25-s − 0.589·26-s − 0.192·27-s − 0.757·29-s + 0.0868·30-s + 1.29·31-s + 0.861·32-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 \( 1 - 0.475T + 2T^{2} \)
13 \( 1 + 6.31T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 3.92T + 19T^{2} \)
23 \( 1 - 4.74T + 23T^{2} \)
29 \( 1 + 4.08T + 29T^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 - 9.04T + 37T^{2} \)
41 \( 1 - 2.67T + 41T^{2} \)
43 \( 1 - 3.43T + 43T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 4.77T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 + 5.93T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 + 6.01T + 71T^{2} \)
73 \( 1 + 6.10T + 73T^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 - 4.23T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−7.61917157123694959675384676139, −6.74224486618789505639980639119, −5.92117581741750983260494751369, −5.28818929271623333435258197385, −4.53986017273306239359107620945, −4.30139520692388679135707403443, −3.14732566190392571119772344088, −2.42438102249079100221900269008, −0.920953741328327670968018603231, 0, 0.920953741328327670968018603231, 2.42438102249079100221900269008, 3.14732566190392571119772344088, 4.30139520692388679135707403443, 4.53986017273306239359107620945, 5.28818929271623333435258197385, 5.92117581741750983260494751369, 6.74224486618789505639980639119, 7.61917157123694959675384676139

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