Properties

Label 2-6026-1.1-c1-0-66
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 + 0.812·3-s + 4-s + 0.413·5-s − 0.812·6-s − 0.531·7-s − 8-s − 2.33·9-s − 0.413·10-s + 5.43·11-s + 0.812·12-s + 0.828·13-s + 0.531·14-s + 0.336·15-s + 16-s + 1.00·17-s + 2.33·18-s − 6.05·19-s + 0.413·20-s − 0.431·21-s − 5.43·22-s + 23-s − 0.812·24-s − 4.82·25-s − 0.828·26-s − 4.34·27-s − 0.531·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.469·3-s + 0.5·4-s + 0.184·5-s − 0.331·6-s − 0.200·7-s − 0.353·8-s − 0.779·9-s − 0.130·10-s + 1.63·11-s + 0.234·12-s + 0.229·13-s + 0.141·14-s + 0.0868·15-s + 0.250·16-s + 0.244·17-s + 0.551·18-s − 1.38·19-s + 0.0924·20-s − 0.0942·21-s − 1.15·22-s + 0.208·23-s − 0.165·24-s − 0.965·25-s − 0.162·26-s − 0.835·27-s − 0.100·28-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

\(L(1)\) \(\approx\) \(1.622977629\)
\(L(\frac12)\) \(\approx\) \(1.622977629\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.812T + 3T^{2} \)
5 \( 1 - 0.413T + 5T^{2} \)
7 \( 1 + 0.531T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 - 0.828T + 13T^{2} \)
17 \( 1 - 1.00T + 17T^{2} \)
19 \( 1 + 6.05T + 19T^{2} \)
29 \( 1 + 0.830T + 29T^{2} \)
31 \( 1 - 3.41T + 31T^{2} \)
37 \( 1 + 4.77T + 37T^{2} \)
41 \( 1 - 8.06T + 41T^{2} \)
43 \( 1 + 0.794T + 43T^{2} \)
47 \( 1 - 8.99T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 4.21T + 59T^{2} \)
61 \( 1 - 4.62T + 61T^{2} \)
67 \( 1 - 2.68T + 67T^{2} \)
71 \( 1 - 7.56T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 1.38T + 79T^{2} \)
83 \( 1 + 2.61T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 - 4.82T + 97T^{2} \)
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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

−8.367063714728764013591123917414, −7.46015578274434672344084497183, −6.65530946529555012728453486676, −6.16930706027663046175243016902, −5.44223798810513339864777574198, −4.11701671169891286318511606762, −3.65795892287988269846685377478, −2.57071363154626288361354426577, −1.86880084368889407156554917377, −0.72524751782374523276315915576, 0.72524751782374523276315915576, 1.86880084368889407156554917377, 2.57071363154626288361354426577, 3.65795892287988269846685377478, 4.11701671169891286318511606762, 5.44223798810513339864777574198, 6.16930706027663046175243016902, 6.65530946529555012728453486676, 7.46015578274434672344084497183, 8.367063714728764013591123917414

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