Properties

Label 2-6026-1.1-c1-0-129
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 + 1.59·3-s + 4-s − 0.438·5-s + 1.59·6-s − 0.361·7-s + 8-s − 0.458·9-s − 0.438·10-s + 4.42·11-s + 1.59·12-s + 4.06·13-s − 0.361·14-s − 0.698·15-s + 16-s + 4.13·17-s − 0.458·18-s + 2.03·19-s − 0.438·20-s − 0.576·21-s + 4.42·22-s + 23-s + 1.59·24-s − 4.80·25-s + 4.06·26-s − 5.51·27-s − 0.361·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.920·3-s + 0.5·4-s − 0.195·5-s + 0.650·6-s − 0.136·7-s + 0.353·8-s − 0.152·9-s − 0.138·10-s + 1.33·11-s + 0.460·12-s + 1.12·13-s − 0.0967·14-s − 0.180·15-s + 0.250·16-s + 1.00·17-s − 0.108·18-s + 0.466·19-s − 0.0979·20-s − 0.125·21-s + 0.942·22-s + 0.208·23-s + 0.325·24-s − 0.961·25-s + 0.796·26-s − 1.06·27-s − 0.0683·28-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

\(L(1)\) \(\approx\) \(4.965454507\)
\(L(\frac12)\) \(\approx\) \(4.965454507\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.59T + 3T^{2} \)
5 \( 1 + 0.438T + 5T^{2} \)
7 \( 1 + 0.361T + 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 - 4.06T + 13T^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
19 \( 1 - 2.03T + 19T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 - 8.21T + 31T^{2} \)
37 \( 1 - 5.31T + 37T^{2} \)
41 \( 1 + 9.41T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 0.939T + 47T^{2} \)
53 \( 1 + 1.19T + 53T^{2} \)
59 \( 1 - 5.01T + 59T^{2} \)
61 \( 1 + 1.04T + 61T^{2} \)
67 \( 1 + 1.18T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 1.86T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 8.29T + 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−8.042357852378170171898382265531, −7.46788493596704472816937265979, −6.50791027908409077886726297892, −5.99447247037841802150881175199, −5.23362807834901937712629511438, −4.06069800930639333593166573737, −3.66608430006776373044834916554, −3.07727303323766120090766069208, −2.01477184853742238094889174307, −1.08182591585893617082808592636, 1.08182591585893617082808592636, 2.01477184853742238094889174307, 3.07727303323766120090766069208, 3.66608430006776373044834916554, 4.06069800930639333593166573737, 5.23362807834901937712629511438, 5.99447247037841802150881175199, 6.50791027908409077886726297892, 7.46788493596704472816937265979, 8.042357852378170171898382265531

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