Properties

Label 2-6026-1.1-c1-0-100
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 + 0.184·3-s + 4-s − 4.21·5-s + 0.184·6-s + 4.87·7-s + 8-s − 2.96·9-s − 4.21·10-s + 5.56·11-s + 0.184·12-s + 2.03·13-s + 4.87·14-s − 0.778·15-s + 16-s + 6.28·17-s − 2.96·18-s + 1.31·19-s − 4.21·20-s + 0.900·21-s + 5.56·22-s + 23-s + 0.184·24-s + 12.7·25-s + 2.03·26-s − 1.10·27-s + 4.87·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.106·3-s + 0.5·4-s − 1.88·5-s + 0.0754·6-s + 1.84·7-s + 0.353·8-s − 0.988·9-s − 1.33·10-s + 1.67·11-s + 0.0533·12-s + 0.564·13-s + 1.30·14-s − 0.200·15-s + 0.250·16-s + 1.52·17-s − 0.699·18-s + 0.301·19-s − 0.941·20-s + 0.196·21-s + 1.18·22-s + 0.208·23-s + 0.0377·24-s + 2.54·25-s + 0.399·26-s − 0.212·27-s + 0.921·28-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

\(L(1)\) \(\approx\) \(3.327348114\)
\(L(\frac12)\) \(\approx\) \(3.327348114\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.184T + 3T^{2} \)
5 \( 1 + 4.21T + 5T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
11 \( 1 - 5.56T + 11T^{2} \)
13 \( 1 - 2.03T + 13T^{2} \)
17 \( 1 - 6.28T + 17T^{2} \)
19 \( 1 - 1.31T + 19T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + 3.22T + 31T^{2} \)
37 \( 1 + 0.993T + 37T^{2} \)
41 \( 1 + 7.61T + 41T^{2} \)
43 \( 1 + 1.71T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 9.76T + 53T^{2} \)
59 \( 1 - 6.87T + 59T^{2} \)
61 \( 1 + 6.10T + 61T^{2} \)
67 \( 1 + 0.0128T + 67T^{2} \)
71 \( 1 - 9.33T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 - 8.45T + 79T^{2} \)
83 \( 1 + 6.31T + 83T^{2} \)
89 \( 1 - 13.9T + 89T^{2} \)
97 \( 1 - 4.91T + 97T^{2} \)
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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

−8.064669129315835797262207965409, −7.45389373187919040551002503811, −6.79738710246397389626174982483, −5.72654634183981147040555716932, −5.05831130067846294567977292810, −4.37848533557863454540263739661, −3.57411641288031198221299847356, −3.33698057011768134979820964907, −1.77081438221176177753747753072, −0.928689981994043817294869608717, 0.928689981994043817294869608717, 1.77081438221176177753747753072, 3.33698057011768134979820964907, 3.57411641288031198221299847356, 4.37848533557863454540263739661, 5.05831130067846294567977292810, 5.72654634183981147040555716932, 6.79738710246397389626174982483, 7.45389373187919040551002503811, 8.064669129315835797262207965409

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