Properties

Label 2-6013-1.1-c1-0-14
Degree $2$
Conductor $6013$
Sign $1$
Analytic cond. $48.0140$
Root an. cond. $6.92921$
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  − 2.11·2-s + 0.0595·3-s + 2.48·4-s − 0.217·5-s − 0.126·6-s + 7-s − 1.02·8-s − 2.99·9-s + 0.460·10-s − 6.48·11-s + 0.147·12-s − 1.76·13-s − 2.11·14-s − 0.0129·15-s − 2.80·16-s − 3.36·17-s + 6.34·18-s + 2.28·19-s − 0.540·20-s + 0.0595·21-s + 13.7·22-s + 4.32·23-s − 0.0608·24-s − 4.95·25-s + 3.73·26-s − 0.357·27-s + 2.48·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 0.0344·3-s + 1.24·4-s − 0.0973·5-s − 0.0515·6-s + 0.377·7-s − 0.361·8-s − 0.998·9-s + 0.145·10-s − 1.95·11-s + 0.0427·12-s − 0.488·13-s − 0.565·14-s − 0.00334·15-s − 0.700·16-s − 0.816·17-s + 1.49·18-s + 0.523·19-s − 0.120·20-s + 0.0130·21-s + 2.92·22-s + 0.900·23-s − 0.0124·24-s − 0.990·25-s + 0.731·26-s − 0.0687·27-s + 0.469·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6013\)    =    \(7 \cdot 859\)
Sign: $1$
Analytic conductor: \(48.0140\)
Root analytic conductor: \(6.92921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6013,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1572481180\)
\(L(\frac12)\) \(\approx\) \(0.1572481180\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
859 \( 1 + T \)
good2 \( 1 + 2.11T + 2T^{2} \)
3 \( 1 - 0.0595T + 3T^{2} \)
5 \( 1 + 0.217T + 5T^{2} \)
11 \( 1 + 6.48T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + 3.36T + 17T^{2} \)
19 \( 1 - 2.28T + 19T^{2} \)
23 \( 1 - 4.32T + 23T^{2} \)
29 \( 1 + 9.07T + 29T^{2} \)
31 \( 1 + 4.78T + 31T^{2} \)
37 \( 1 + 5.25T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 9.64T + 43T^{2} \)
47 \( 1 + 1.63T + 47T^{2} \)
53 \( 1 - 1.70T + 53T^{2} \)
59 \( 1 + 2.35T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 + 5.08T + 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 7.17T + 79T^{2} \)
83 \( 1 - 3.43T + 83T^{2} \)
89 \( 1 - 1.17T + 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−8.076782407426876725993803388749, −7.54643412006628726426308334676, −7.21109853110255974521710501299, −5.96888944775813360385076811976, −5.31994466641040691420309346941, −4.62871887592744923534326062423, −3.28932502778766110006342952567, −2.45535507139727986541488557856, −1.78257980565297683983528232550, −0.24646800945357377149862680335, 0.24646800945357377149862680335, 1.78257980565297683983528232550, 2.45535507139727986541488557856, 3.28932502778766110006342952567, 4.62871887592744923534326062423, 5.31994466641040691420309346941, 5.96888944775813360385076811976, 7.21109853110255974521710501299, 7.54643412006628726426308334676, 8.076782407426876725993803388749

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