Properties

Label 2-8030-1.1-c1-0-206
Degree $2$
Conductor $8030$
Sign $-1$
Analytic cond. $64.1198$
Root an. cond. $8.00748$
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-s + 2.89·3-s + 4-s + 5-s − 2.89·6-s − 4.53·7-s − 8-s + 5.39·9-s − 10-s − 11-s + 2.89·12-s − 1.25·13-s + 4.53·14-s + 2.89·15-s + 16-s − 3.36·17-s − 5.39·18-s + 0.294·19-s + 20-s − 13.1·21-s + 22-s + 0.544·23-s − 2.89·24-s + 25-s + 1.25·26-s + 6.94·27-s − 4.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.67·3-s + 0.5·4-s + 0.447·5-s − 1.18·6-s − 1.71·7-s − 0.353·8-s + 1.79·9-s − 0.316·10-s − 0.301·11-s + 0.836·12-s − 0.347·13-s + 1.21·14-s + 0.748·15-s + 0.250·16-s − 0.815·17-s − 1.27·18-s + 0.0676·19-s + 0.223·20-s − 2.86·21-s + 0.213·22-s + 0.113·23-s − 0.591·24-s + 0.200·25-s + 0.245·26-s + 1.33·27-s − 0.857·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8030\)    =    \(2 \cdot 5 \cdot 11 \cdot 73\)
Sign: $-1$
Analytic conductor: \(64.1198\)
Root analytic conductor: \(8.00748\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8030,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
73 \( 1 - T \)
good3 \( 1 - 2.89T + 3T^{2} \)
7 \( 1 + 4.53T + 7T^{2} \)
13 \( 1 + 1.25T + 13T^{2} \)
17 \( 1 + 3.36T + 17T^{2} \)
19 \( 1 - 0.294T + 19T^{2} \)
23 \( 1 - 0.544T + 23T^{2} \)
29 \( 1 - 4.46T + 29T^{2} \)
31 \( 1 + 0.755T + 31T^{2} \)
37 \( 1 - 0.0293T + 37T^{2} \)
41 \( 1 - 4.70T + 41T^{2} \)
43 \( 1 + 4.20T + 43T^{2} \)
47 \( 1 - 7.94T + 47T^{2} \)
53 \( 1 + 9.14T + 53T^{2} \)
59 \( 1 + 15.0T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
79 \( 1 - 8.72T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 9.24T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.72584411530660162593404853038, −6.74079466006147755657315435814, −6.61687576970808726940660090006, −5.55311681645878693389952322776, −4.38148377161759718327723300451, −3.57099226972241876280421819041, −2.73480278729292401393293976526, −2.57626756906230973422251532018, −1.43153014026244687216725012174, 0, 1.43153014026244687216725012174, 2.57626756906230973422251532018, 2.73480278729292401393293976526, 3.57099226972241876280421819041, 4.38148377161759718327723300451, 5.55311681645878693389952322776, 6.61687576970808726940660090006, 6.74079466006147755657315435814, 7.72584411530660162593404853038

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