Properties

Label 2-8085-1.1-c1-0-64
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  + 0.414·2-s + 3-s − 1.82·4-s + 5-s + 0.414·6-s − 1.58·8-s + 9-s + 0.414·10-s − 11-s − 1.82·12-s − 5.65·13-s + 15-s + 3·16-s + 6.82·17-s + 0.414·18-s + 1.17·19-s − 1.82·20-s − 0.414·22-s − 4·23-s − 1.58·24-s + 25-s − 2.34·26-s + 27-s + 0.828·29-s + 0.414·30-s + 4.41·32-s − 33-s + ⋯
L(s)  = 1  + 0.292·2-s + 0.577·3-s − 0.914·4-s + 0.447·5-s + 0.169·6-s − 0.560·8-s + 0.333·9-s + 0.130·10-s − 0.301·11-s − 0.527·12-s − 1.56·13-s + 0.258·15-s + 0.750·16-s + 1.65·17-s + 0.0976·18-s + 0.268·19-s − 0.408·20-s − 0.0883·22-s − 0.834·23-s − 0.323·24-s + 0.200·25-s − 0.459·26-s + 0.192·27-s + 0.153·29-s + 0.0756·30-s + 0.780·32-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

\(L(1)\) \(\approx\) \(2.158464196\)
\(L(\frac12)\) \(\approx\) \(2.158464196\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414T + 2T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 0.343T + 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 + 3.17T + 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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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

−7.962804422537913427106966580640, −7.30165974489213326592700589986, −6.37275172729579456124721431034, −5.45060156648847116531990534536, −5.12087331134951538378229128877, −4.30638547733632780092568033925, −3.46277127068768429339540672212, −2.82536210325336880229873980945, −1.89866558164410031499953653343, −0.67295794320558912994865498889, 0.67295794320558912994865498889, 1.89866558164410031499953653343, 2.82536210325336880229873980945, 3.46277127068768429339540672212, 4.30638547733632780092568033925, 5.12087331134951538378229128877, 5.45060156648847116531990534536, 6.37275172729579456124721431034, 7.30165974489213326592700589986, 7.962804422537913427106966580640

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