Properties

Label 2-6026-1.1-c1-0-43
Degree $2$
Conductor $6026$
Sign $1$
Analytic cond. $48.1178$
Root an. cond. $6.93670$
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  − 2-s − 1.47·3-s + 4-s + 2.55·5-s + 1.47·6-s − 3.54·7-s − 8-s − 0.817·9-s − 2.55·10-s − 0.373·11-s − 1.47·12-s − 2.41·13-s + 3.54·14-s − 3.77·15-s + 16-s − 0.178·17-s + 0.817·18-s + 6.41·19-s + 2.55·20-s + 5.23·21-s + 0.373·22-s + 23-s + 1.47·24-s + 1.51·25-s + 2.41·26-s + 5.63·27-s − 3.54·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.852·3-s + 0.5·4-s + 1.14·5-s + 0.603·6-s − 1.33·7-s − 0.353·8-s − 0.272·9-s − 0.807·10-s − 0.112·11-s − 0.426·12-s − 0.668·13-s + 0.946·14-s − 0.973·15-s + 0.250·16-s − 0.0431·17-s + 0.192·18-s + 1.47·19-s + 0.570·20-s + 1.14·21-s + 0.0796·22-s + 0.208·23-s + 0.301·24-s + 0.303·25-s + 0.472·26-s + 1.08·27-s − 0.669·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6026\)    =    \(2 \cdot 23 \cdot 131\)
Sign: $1$
Analytic conductor: \(48.1178\)
Root analytic conductor: \(6.93670\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7398359458\)
\(L(\frac12)\) \(\approx\) \(0.7398359458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 - T \)
131 \( 1 + T \)
good3 \( 1 + 1.47T + 3T^{2} \)
5 \( 1 - 2.55T + 5T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 + 0.373T + 11T^{2} \)
13 \( 1 + 2.41T + 13T^{2} \)
17 \( 1 + 0.178T + 17T^{2} \)
19 \( 1 - 6.41T + 19T^{2} \)
29 \( 1 - 9.16T + 29T^{2} \)
31 \( 1 - 4.56T + 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 + 9.99T + 41T^{2} \)
43 \( 1 + 6.55T + 43T^{2} \)
47 \( 1 + 8.10T + 47T^{2} \)
53 \( 1 + 3.45T + 53T^{2} \)
59 \( 1 + 6.29T + 59T^{2} \)
61 \( 1 + 1.13T + 61T^{2} \)
67 \( 1 + 3.19T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + 15.4T + 73T^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 - 4.76T + 83T^{2} \)
89 \( 1 + 7.78T + 89T^{2} \)
97 \( 1 - 1.11T + 97T^{2} \)
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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

−8.170550756515625890049637217020, −7.15884073044916516583872951463, −6.48708523224400440104104223308, −6.21417764921899076266041673059, −5.34254089731495820488428302432, −4.83126514282401873979876627992, −3.17577442412494463591181071610, −2.85968672763566731821537172661, −1.63223523923147217851837805423, −0.52262054811226271474293860907, 0.52262054811226271474293860907, 1.63223523923147217851837805423, 2.85968672763566731821537172661, 3.17577442412494463591181071610, 4.83126514282401873979876627992, 5.34254089731495820488428302432, 6.21417764921899076266041673059, 6.48708523224400440104104223308, 7.15884073044916516583872951463, 8.170550756515625890049637217020

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