Properties

Label 2-6026-1.1-c1-0-127
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.37·3-s + 4-s − 2.07·5-s − 2.37·6-s − 0.108·7-s + 8-s + 2.64·9-s − 2.07·10-s − 2.80·11-s − 2.37·12-s + 0.0828·13-s − 0.108·14-s + 4.93·15-s + 16-s − 0.866·17-s + 2.64·18-s + 2.95·19-s − 2.07·20-s + 0.257·21-s − 2.80·22-s − 23-s − 2.37·24-s − 0.686·25-s + 0.0828·26-s + 0.834·27-s − 0.108·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.37·3-s + 0.5·4-s − 0.928·5-s − 0.970·6-s − 0.0408·7-s + 0.353·8-s + 0.882·9-s − 0.656·10-s − 0.845·11-s − 0.686·12-s + 0.0229·13-s − 0.0289·14-s + 1.27·15-s + 0.250·16-s − 0.210·17-s + 0.624·18-s + 0.677·19-s − 0.464·20-s + 0.0561·21-s − 0.598·22-s − 0.208·23-s − 0.485·24-s − 0.137·25-s + 0.0162·26-s + 0.160·27-s − 0.0204·28-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: \(1\)
Selberg data: \((2,\ 6026,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
23 \( 1 + T \)
131 \( 1 - T \)
good3 \( 1 + 2.37T + 3T^{2} \)
5 \( 1 + 2.07T + 5T^{2} \)
7 \( 1 + 0.108T + 7T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 - 0.0828T + 13T^{2} \)
17 \( 1 + 0.866T + 17T^{2} \)
19 \( 1 - 2.95T + 19T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 5.07T + 31T^{2} \)
37 \( 1 - 7.73T + 37T^{2} \)
41 \( 1 - 6.79T + 41T^{2} \)
43 \( 1 - 3.52T + 43T^{2} \)
47 \( 1 + 2.96T + 47T^{2} \)
53 \( 1 - 5.25T + 53T^{2} \)
59 \( 1 + 2.25T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 + 3.89T + 71T^{2} \)
73 \( 1 + 9.27T + 73T^{2} \)
79 \( 1 - 7.76T + 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + 5.48T + 97T^{2} \)
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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

−7.68705578344328183862158024565, −6.84257155843597397040128407108, −6.07846482760223864546532059808, −5.64755147366990097011886596243, −4.69877037704394451091583030944, −4.42897471877567395836975045376, −3.35594992862249814660446995865, −2.52623036831299463284025410069, −1.07573711363988427018462096632, 0, 1.07573711363988427018462096632, 2.52623036831299463284025410069, 3.35594992862249814660446995865, 4.42897471877567395836975045376, 4.69877037704394451091583030944, 5.64755147366990097011886596243, 6.07846482760223864546532059808, 6.84257155843597397040128407108, 7.68705578344328183862158024565

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