Properties

Label 2-6026-1.1-c1-0-160
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 + 2.60·3-s + 4-s − 1.20·5-s − 2.60·6-s + 4.34·7-s − 8-s + 3.77·9-s + 1.20·10-s + 4.29·11-s + 2.60·12-s + 2.62·13-s − 4.34·14-s − 3.12·15-s + 16-s + 6.73·17-s − 3.77·18-s + 3.92·19-s − 1.20·20-s + 11.3·21-s − 4.29·22-s + 23-s − 2.60·24-s − 3.55·25-s − 2.62·26-s + 2.00·27-s + 4.34·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.50·3-s + 0.5·4-s − 0.537·5-s − 1.06·6-s + 1.64·7-s − 0.353·8-s + 1.25·9-s + 0.379·10-s + 1.29·11-s + 0.751·12-s + 0.727·13-s − 1.16·14-s − 0.807·15-s + 0.250·16-s + 1.63·17-s − 0.888·18-s + 0.901·19-s − 0.268·20-s + 2.46·21-s − 0.916·22-s + 0.208·23-s − 0.531·24-s − 0.711·25-s − 0.514·26-s + 0.386·27-s + 0.821·28-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

\(L(1)\) \(\approx\) \(3.560851182\)
\(L(\frac12)\) \(\approx\) \(3.560851182\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 + 1.20T + 5T^{2} \)
7 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 - 4.29T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 6.73T + 17T^{2} \)
19 \( 1 - 3.92T + 19T^{2} \)
29 \( 1 + 2.61T + 29T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 + 4.57T + 37T^{2} \)
41 \( 1 - 2.83T + 41T^{2} \)
43 \( 1 - 7.42T + 43T^{2} \)
47 \( 1 - 4.72T + 47T^{2} \)
53 \( 1 + 1.15T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 0.687T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 + 7.47T + 79T^{2} \)
83 \( 1 - 5.43T + 83T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−8.093666975072689600940157024626, −7.56321849712796225478575913439, −7.30231197361569548238768379684, −5.99801724338404198429254359318, −5.20231100567691997988617389795, −3.95840773746027673405497017957, −3.71988201516830599870062459964, −2.70988246675683450284801044985, −1.56794958557665161785492460173, −1.25603152112178430048344015991, 1.25603152112178430048344015991, 1.56794958557665161785492460173, 2.70988246675683450284801044985, 3.71988201516830599870062459964, 3.95840773746027673405497017957, 5.20231100567691997988617389795, 5.99801724338404198429254359318, 7.30231197361569548238768379684, 7.56321849712796225478575913439, 8.093666975072689600940157024626

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