Properties

Degree $2$
Conductor $6001$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·2-s + 1.01·3-s + 4.58·4-s + 2.09·5-s − 2.59·6-s + 3.20·7-s − 6.63·8-s − 1.97·9-s − 5.38·10-s − 3.82·11-s + 4.63·12-s + 3.94·13-s − 8.21·14-s + 2.12·15-s + 7.84·16-s + 17-s + 5.07·18-s − 1.67·19-s + 9.62·20-s + 3.24·21-s + 9.80·22-s − 6.51·23-s − 6.70·24-s − 0.595·25-s − 10.1·26-s − 5.03·27-s + 14.6·28-s + ⋯
L(s)  = 1  − 1.81·2-s + 0.584·3-s + 2.29·4-s + 0.938·5-s − 1.05·6-s + 1.21·7-s − 2.34·8-s − 0.658·9-s − 1.70·10-s − 1.15·11-s + 1.33·12-s + 1.09·13-s − 2.19·14-s + 0.548·15-s + 1.96·16-s + 0.242·17-s + 1.19·18-s − 0.383·19-s + 2.15·20-s + 0.707·21-s + 2.09·22-s − 1.35·23-s − 1.36·24-s − 0.119·25-s − 1.98·26-s − 0.968·27-s + 2.77·28-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{6001} (1, \cdot )$
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 \( 1 + 2.56T + 2T^{2} \)
3 \( 1 - 1.01T + 3T^{2} \)
5 \( 1 - 2.09T + 5T^{2} \)
7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 + 3.82T + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
19 \( 1 + 1.67T + 19T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 + 7.47T + 29T^{2} \)
31 \( 1 - 5.60T + 31T^{2} \)
37 \( 1 + 0.965T + 37T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
43 \( 1 + 5.00T + 43T^{2} \)
47 \( 1 + 8.53T + 47T^{2} \)
53 \( 1 + 1.22T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 4.75T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 0.894T + 71T^{2} \)
73 \( 1 - 2.09T + 73T^{2} \)
79 \( 1 + 8.36T + 79T^{2} \)
83 \( 1 + 9.04T + 83T^{2} \)
89 \( 1 - 1.32T + 89T^{2} \)
97 \( 1 + 0.597T + 97T^{2} \)
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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

−8.125636749737312866539494509169, −7.49836852736322929366934102791, −6.42937948585524052552651534138, −5.82748869530338377447932452684, −5.14036094688500247421536008224, −3.74312568339314961886118895561, −2.64438901186438103036400502827, −2.01690080577759342682271958808, −1.45445859689579845516297193052, 0, 1.45445859689579845516297193052, 2.01690080577759342682271958808, 2.64438901186438103036400502827, 3.74312568339314961886118895561, 5.14036094688500247421536008224, 5.82748869530338377447932452684, 6.42937948585524052552651534138, 7.49836852736322929366934102791, 8.125636749737312866539494509169

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