Properties

Label 2-8047-1.1-c1-0-427
Degree $2$
Conductor $8047$
Sign $-1$
Analytic cond. $64.2556$
Root an. cond. $8.01596$
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  − 2.23·2-s − 0.938·3-s + 3.00·4-s + 3.28·5-s + 2.09·6-s + 1.35·7-s − 2.25·8-s − 2.12·9-s − 7.35·10-s − 0.595·11-s − 2.81·12-s + 13-s − 3.02·14-s − 3.08·15-s − 0.977·16-s + 5.30·17-s + 4.74·18-s − 1.67·19-s + 9.88·20-s − 1.26·21-s + 1.33·22-s − 8.87·23-s + 2.11·24-s + 5.82·25-s − 2.23·26-s + 4.80·27-s + 4.05·28-s + ⋯
L(s)  = 1  − 1.58·2-s − 0.541·3-s + 1.50·4-s + 1.47·5-s + 0.856·6-s + 0.510·7-s − 0.795·8-s − 0.706·9-s − 2.32·10-s − 0.179·11-s − 0.813·12-s + 0.277·13-s − 0.807·14-s − 0.796·15-s − 0.244·16-s + 1.28·17-s + 1.11·18-s − 0.383·19-s + 2.21·20-s − 0.276·21-s + 0.283·22-s − 1.85·23-s + 0.430·24-s + 1.16·25-s − 0.438·26-s + 0.924·27-s + 0.767·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8047\)    =    \(13 \cdot 619\)
Sign: $-1$
Analytic conductor: \(64.2556\)
Root analytic conductor: \(8.01596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8047,\ (\ :1/2),\ -1)\)

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)$
bad13 \( 1 - T \)
619 \( 1 - T \)
good2 \( 1 + 2.23T + 2T^{2} \)
3 \( 1 + 0.938T + 3T^{2} \)
5 \( 1 - 3.28T + 5T^{2} \)
7 \( 1 - 1.35T + 7T^{2} \)
11 \( 1 + 0.595T + 11T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 + 1.67T + 19T^{2} \)
23 \( 1 + 8.87T + 23T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
31 \( 1 + 2.30T + 31T^{2} \)
37 \( 1 + 4.70T + 37T^{2} \)
41 \( 1 + 4.26T + 41T^{2} \)
43 \( 1 + 2.71T + 43T^{2} \)
47 \( 1 - 5.75T + 47T^{2} \)
53 \( 1 - 6.39T + 53T^{2} \)
59 \( 1 - 2.32T + 59T^{2} \)
61 \( 1 - 0.743T + 61T^{2} \)
67 \( 1 - 4.96T + 67T^{2} \)
71 \( 1 + 0.701T + 71T^{2} \)
73 \( 1 - 7.63T + 73T^{2} \)
79 \( 1 - 3.41T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + 9.76T + 97T^{2} \)
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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

−7.74054179805625008256761132714, −6.81327497417813252754637033702, −6.18912396030929880817896284268, −5.58718884231707420711196761111, −5.08754783339330518919740802421, −3.77450333867015383982134722985, −2.55715953920761189813981011710, −1.90451167548281186176283841854, −1.19260431567102767978380735901, 0, 1.19260431567102767978380735901, 1.90451167548281186176283841854, 2.55715953920761189813981011710, 3.77450333867015383982134722985, 5.08754783339330518919740802421, 5.58718884231707420711196761111, 6.18912396030929880817896284268, 6.81327497417813252754637033702, 7.74054179805625008256761132714

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