Properties

Label 2-8015-1.1-c1-0-39
Degree $2$
Conductor $8015$
Sign $1$
Analytic cond. $64.0000$
Root an. cond. $8.00000$
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  + 1.36·2-s − 0.257·3-s − 0.148·4-s − 5-s − 0.349·6-s − 7-s − 2.92·8-s − 2.93·9-s − 1.36·10-s + 2.45·11-s + 0.0381·12-s − 6.54·13-s − 1.36·14-s + 0.257·15-s − 3.68·16-s + 2.41·17-s − 3.99·18-s − 1.82·19-s + 0.148·20-s + 0.257·21-s + 3.33·22-s − 3.40·23-s + 0.751·24-s + 25-s − 8.90·26-s + 1.52·27-s + 0.148·28-s + ⋯
L(s)  = 1  + 0.962·2-s − 0.148·3-s − 0.0741·4-s − 0.447·5-s − 0.142·6-s − 0.377·7-s − 1.03·8-s − 0.977·9-s − 0.430·10-s + 0.739·11-s + 0.0110·12-s − 1.81·13-s − 0.363·14-s + 0.0663·15-s − 0.920·16-s + 0.585·17-s − 0.940·18-s − 0.419·19-s + 0.0331·20-s + 0.0561·21-s + 0.711·22-s − 0.709·23-s + 0.153·24-s + 0.200·25-s − 1.74·26-s + 0.293·27-s + 0.0280·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8015\)    =    \(5 \cdot 7 \cdot 229\)
Sign: $1$
Analytic conductor: \(64.0000\)
Root analytic conductor: \(8.00000\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8015,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8126837733\)
\(L(\frac12)\) \(\approx\) \(0.8126837733\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 + T \)
229 \( 1 - T \)
good2 \( 1 - 1.36T + 2T^{2} \)
3 \( 1 + 0.257T + 3T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 + 6.54T + 13T^{2} \)
17 \( 1 - 2.41T + 17T^{2} \)
19 \( 1 + 1.82T + 19T^{2} \)
23 \( 1 + 3.40T + 23T^{2} \)
29 \( 1 + 6.44T + 29T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 + 4.22T + 37T^{2} \)
41 \( 1 - 2.29T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 - 2.83T + 53T^{2} \)
59 \( 1 + 2.81T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 6.06T + 67T^{2} \)
71 \( 1 - 5.47T + 71T^{2} \)
73 \( 1 - 4.63T + 73T^{2} \)
79 \( 1 + 0.283T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−7.75458686802721874579638333215, −6.98284797752945281776622131178, −6.32088337990853030723842293200, −5.47194584828473828774753216803, −5.18537500072603811336144367962, −4.19690409232893156971135345500, −3.68120259483118043363184524868, −2.91176953050943736601067836510, −2.09585891272330349586040579338, −0.36226164702626314854019092635, 0.36226164702626314854019092635, 2.09585891272330349586040579338, 2.91176953050943736601067836510, 3.68120259483118043363184524868, 4.19690409232893156971135345500, 5.18537500072603811336144367962, 5.47194584828473828774753216803, 6.32088337990853030723842293200, 6.98284797752945281776622131178, 7.75458686802721874579638333215

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