Properties

Label 2-6026-1.1-c1-0-153
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 − 1.87·3-s + 4-s + 0.417·5-s − 1.87·6-s − 3.88·7-s + 8-s + 0.514·9-s + 0.417·10-s + 1.68·11-s − 1.87·12-s + 0.977·13-s − 3.88·14-s − 0.782·15-s + 16-s + 4.13·17-s + 0.514·18-s + 0.960·19-s + 0.417·20-s + 7.28·21-s + 1.68·22-s − 23-s − 1.87·24-s − 4.82·25-s + 0.977·26-s + 4.65·27-s − 3.88·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.08·3-s + 0.5·4-s + 0.186·5-s − 0.765·6-s − 1.46·7-s + 0.353·8-s + 0.171·9-s + 0.131·10-s + 0.508·11-s − 0.541·12-s + 0.271·13-s − 1.03·14-s − 0.201·15-s + 0.250·16-s + 1.00·17-s + 0.121·18-s + 0.220·19-s + 0.0932·20-s + 1.58·21-s + 0.359·22-s − 0.208·23-s − 0.382·24-s − 0.965·25-s + 0.191·26-s + 0.896·27-s − 0.734·28-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 \( 1 + 1.87T + 3T^{2} \)
5 \( 1 - 0.417T + 5T^{2} \)
7 \( 1 + 3.88T + 7T^{2} \)
11 \( 1 - 1.68T + 11T^{2} \)
13 \( 1 - 0.977T + 13T^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
19 \( 1 - 0.960T + 19T^{2} \)
29 \( 1 - 2.28T + 29T^{2} \)
31 \( 1 + 5.85T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 - 5.25T + 43T^{2} \)
47 \( 1 - 9.97T + 47T^{2} \)
53 \( 1 + 7.30T + 53T^{2} \)
59 \( 1 - 2.12T + 59T^{2} \)
61 \( 1 - 4.46T + 61T^{2} \)
67 \( 1 + 3.65T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 1.30T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 2.72T + 83T^{2} \)
89 \( 1 - 3.02T + 89T^{2} \)
97 \( 1 + 7.36T + 97T^{2} \)
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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

−7.39914619508906832602932437984, −6.64262362617490786886144608791, −6.25608448989188431111522789401, −5.58449512653730312607971106786, −5.13285553245779407727841735759, −3.89786054815545765111285828997, −3.48759485156334037676931340640, −2.50606952528396507056127425044, −1.21625359395333293795086999634, 0, 1.21625359395333293795086999634, 2.50606952528396507056127425044, 3.48759485156334037676931340640, 3.89786054815545765111285828997, 5.13285553245779407727841735759, 5.58449512653730312607971106786, 6.25608448989188431111522789401, 6.64262362617490786886144608791, 7.39914619508906832602932437984

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