Properties

Label 2-6026-1.1-c1-0-179
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 + 3.19·3-s + 4-s + 3.71·5-s − 3.19·6-s + 5.02·7-s − 8-s + 7.23·9-s − 3.71·10-s − 5.74·11-s + 3.19·12-s − 4.82·13-s − 5.02·14-s + 11.8·15-s + 16-s + 4.96·17-s − 7.23·18-s + 2.21·19-s + 3.71·20-s + 16.0·21-s + 5.74·22-s − 23-s − 3.19·24-s + 8.81·25-s + 4.82·26-s + 13.5·27-s + 5.02·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.84·3-s + 0.5·4-s + 1.66·5-s − 1.30·6-s + 1.89·7-s − 0.353·8-s + 2.41·9-s − 1.17·10-s − 1.73·11-s + 0.923·12-s − 1.33·13-s − 1.34·14-s + 3.07·15-s + 0.250·16-s + 1.20·17-s − 1.70·18-s + 0.508·19-s + 0.830·20-s + 3.50·21-s + 1.22·22-s − 0.208·23-s − 0.653·24-s + 1.76·25-s + 0.945·26-s + 2.61·27-s + 0.949·28-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

\(L(1)\) \(\approx\) \(4.841258341\)
\(L(\frac12)\) \(\approx\) \(4.841258341\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.19T + 3T^{2} \)
5 \( 1 - 3.71T + 5T^{2} \)
7 \( 1 - 5.02T + 7T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 - 2.21T + 19T^{2} \)
29 \( 1 + 4.89T + 29T^{2} \)
31 \( 1 - 6.67T + 31T^{2} \)
37 \( 1 + 0.666T + 37T^{2} \)
41 \( 1 - 1.60T + 41T^{2} \)
43 \( 1 - 4.96T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 2.18T + 59T^{2} \)
61 \( 1 + 4.33T + 61T^{2} \)
67 \( 1 - 5.89T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 4.35T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 5.52T + 83T^{2} \)
89 \( 1 - 0.677T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−8.024231824291439767880708273670, −7.70256731889054946681800279049, −7.22554425242755801566953097898, −5.86854368316075238780682871245, −5.09994681923682626247623549871, −4.63896105472667783944340208523, −3.05437186936783982083243705978, −2.56171680593067153061481225258, −1.90886075566929730030173401988, −1.37215432260386406030972238614, 1.37215432260386406030972238614, 1.90886075566929730030173401988, 2.56171680593067153061481225258, 3.05437186936783982083243705978, 4.63896105472667783944340208523, 5.09994681923682626247623549871, 5.86854368316075238780682871245, 7.22554425242755801566953097898, 7.70256731889054946681800279049, 8.024231824291439767880708273670

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