Properties

Label 2-6001-1.1-c1-0-153
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.36·2-s − 1.16·3-s + 3.59·4-s − 3.78·5-s + 2.74·6-s − 1.24·7-s − 3.78·8-s − 1.65·9-s + 8.95·10-s + 5.90·11-s − 4.17·12-s − 1.18·13-s + 2.95·14-s + 4.39·15-s + 1.75·16-s + 17-s + 3.90·18-s − 1.27·19-s − 13.6·20-s + 1.45·21-s − 13.9·22-s − 7.24·23-s + 4.39·24-s + 9.32·25-s + 2.79·26-s + 5.40·27-s − 4.49·28-s + ⋯
L(s)  = 1  − 1.67·2-s − 0.670·3-s + 1.79·4-s − 1.69·5-s + 1.12·6-s − 0.472·7-s − 1.33·8-s − 0.550·9-s + 2.83·10-s + 1.77·11-s − 1.20·12-s − 0.327·13-s + 0.790·14-s + 1.13·15-s + 0.438·16-s + 0.242·17-s + 0.921·18-s − 0.291·19-s − 3.04·20-s + 0.316·21-s − 2.97·22-s − 1.51·23-s + 0.896·24-s + 1.86·25-s + 0.548·26-s + 1.03·27-s − 0.849·28-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: \(1\)
Selberg data: \((2,\ 6001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.36T + 2T^{2} \)
3 \( 1 + 1.16T + 3T^{2} \)
5 \( 1 + 3.78T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 - 5.90T + 11T^{2} \)
13 \( 1 + 1.18T + 13T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
23 \( 1 + 7.24T + 23T^{2} \)
29 \( 1 + 9.12T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 + 0.876T + 43T^{2} \)
47 \( 1 + 3.03T + 47T^{2} \)
53 \( 1 - 6.64T + 53T^{2} \)
59 \( 1 - 2.21T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 4.23T + 67T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 + 7.52T + 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 4.50T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.72026193253079980550471923928, −7.35467687237034528070544681717, −6.40291063744416953801127517175, −6.14408772215701111092214461057, −4.72418948328728175134330133148, −3.91683688837424391376299189315, −3.22116660574461081415219801851, −1.90638218151000654122743634472, −0.74975291948737585402022018519, 0, 0.74975291948737585402022018519, 1.90638218151000654122743634472, 3.22116660574461081415219801851, 3.91683688837424391376299189315, 4.72418948328728175134330133148, 6.14408772215701111092214461057, 6.40291063744416953801127517175, 7.35467687237034528070544681717, 7.72026193253079980550471923928

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