Properties

Label 2-6026-1.1-c1-0-48
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 − 1.69·3-s + 4-s − 3.34·5-s − 1.69·6-s + 0.398·7-s + 8-s − 0.140·9-s − 3.34·10-s + 2.98·11-s − 1.69·12-s + 4.90·13-s + 0.398·14-s + 5.65·15-s + 16-s − 7.88·17-s − 0.140·18-s + 7.38·19-s − 3.34·20-s − 0.674·21-s + 2.98·22-s − 23-s − 1.69·24-s + 6.19·25-s + 4.90·26-s + 5.31·27-s + 0.398·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.976·3-s + 0.5·4-s − 1.49·5-s − 0.690·6-s + 0.150·7-s + 0.353·8-s − 0.0467·9-s − 1.05·10-s + 0.898·11-s − 0.488·12-s + 1.35·13-s + 0.106·14-s + 1.46·15-s + 0.250·16-s − 1.91·17-s − 0.0330·18-s + 1.69·19-s − 0.748·20-s − 0.147·21-s + 0.635·22-s − 0.208·23-s − 0.345·24-s + 1.23·25-s + 0.961·26-s + 1.02·27-s + 0.0753·28-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

\(L(1)\) \(\approx\) \(1.402317211\)
\(L(\frac12)\) \(\approx\) \(1.402317211\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69T + 3T^{2} \)
5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 - 0.398T + 7T^{2} \)
11 \( 1 - 2.98T + 11T^{2} \)
13 \( 1 - 4.90T + 13T^{2} \)
17 \( 1 + 7.88T + 17T^{2} \)
19 \( 1 - 7.38T + 19T^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 + 8.61T + 37T^{2} \)
41 \( 1 + 6.81T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 2.44T + 47T^{2} \)
53 \( 1 - 9.95T + 53T^{2} \)
59 \( 1 + 2.60T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 - 8.52T + 71T^{2} \)
73 \( 1 - 2.01T + 73T^{2} \)
79 \( 1 - 6.10T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 2.06T + 97T^{2} \)
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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

−7.949247711837089913696960267848, −7.05849931680260030434638671594, −6.70008868852340112527550607402, −5.86550331968046599117646025349, −5.17697702448869494590459012559, −4.41150058779545837954659802219, −3.75049518051160805853179879008, −3.23498073416861488181426116018, −1.74331588587805916131298773136, −0.59447199728121113332842566446, 0.59447199728121113332842566446, 1.74331588587805916131298773136, 3.23498073416861488181426116018, 3.75049518051160805853179879008, 4.41150058779545837954659802219, 5.17697702448869494590459012559, 5.86550331968046599117646025349, 6.70008868852340112527550607402, 7.05849931680260030434638671594, 7.949247711837089913696960267848

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