Properties

Label 2-6001-1.1-c1-0-52
Degree $2$
Conductor $6001$
Sign $1$
Analytic cond. $47.9182$
Root an. cond. $6.92229$
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  − 2.25·2-s − 3.40·3-s + 3.07·4-s − 2.33·5-s + 7.66·6-s + 2.81·7-s − 2.41·8-s + 8.58·9-s + 5.26·10-s + 1.91·11-s − 10.4·12-s − 4.71·13-s − 6.34·14-s + 7.96·15-s − 0.700·16-s + 17-s − 19.3·18-s − 2.11·19-s − 7.19·20-s − 9.58·21-s − 4.32·22-s − 5.88·23-s + 8.23·24-s + 0.473·25-s + 10.6·26-s − 19.0·27-s + 8.65·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 1.96·3-s + 1.53·4-s − 1.04·5-s + 3.12·6-s + 1.06·7-s − 0.854·8-s + 2.86·9-s + 1.66·10-s + 0.578·11-s − 3.01·12-s − 1.30·13-s − 1.69·14-s + 2.05·15-s − 0.175·16-s + 0.242·17-s − 4.55·18-s − 0.485·19-s − 1.60·20-s − 2.09·21-s − 0.921·22-s − 1.22·23-s + 1.67·24-s + 0.0946·25-s + 2.08·26-s − 3.65·27-s + 1.63·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $1$
Analytic conductor: \(47.9182\)
Root analytic conductor: \(6.92229\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2114461908\)
\(L(\frac12)\) \(\approx\) \(0.2114461908\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
353 \( 1 + T \)
good2 \( 1 + 2.25T + 2T^{2} \)
3 \( 1 + 3.40T + 3T^{2} \)
5 \( 1 + 2.33T + 5T^{2} \)
7 \( 1 - 2.81T + 7T^{2} \)
11 \( 1 - 1.91T + 11T^{2} \)
13 \( 1 + 4.71T + 13T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 + 5.88T + 23T^{2} \)
29 \( 1 - 6.32T + 29T^{2} \)
31 \( 1 - 6.02T + 31T^{2} \)
37 \( 1 + 6.28T + 37T^{2} \)
41 \( 1 - 1.40T + 41T^{2} \)
43 \( 1 + 4.35T + 43T^{2} \)
47 \( 1 - 0.504T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 - 5.35T + 59T^{2} \)
61 \( 1 - 7.70T + 61T^{2} \)
67 \( 1 - 6.86T + 67T^{2} \)
71 \( 1 - 0.718T + 71T^{2} \)
73 \( 1 + 6.38T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 - 5.28T + 89T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−7.927599247371859467672654535272, −7.52873775572306358403479353042, −6.78370769432948536132918409772, −6.29860588082621009036230485943, −5.15866928589183062958336179026, −4.63947674670027757351256749988, −3.96931620225093770814376773859, −2.15381657434284400857396640853, −1.27919888892212016567433500919, −0.38888147486225929137697141496, 0.38888147486225929137697141496, 1.27919888892212016567433500919, 2.15381657434284400857396640853, 3.96931620225093770814376773859, 4.63947674670027757351256749988, 5.15866928589183062958336179026, 6.29860588082621009036230485943, 6.78370769432948536132918409772, 7.52873775572306358403479353042, 7.927599247371859467672654535272

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