Properties

Label 2-6003-1.1-c1-0-180
Degree $2$
Conductor $6003$
Sign $-1$
Analytic cond. $47.9341$
Root an. cond. $6.92345$
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  − 2.08·2-s + 2.36·4-s − 0.182·5-s + 1.65·7-s − 0.758·8-s + 0.380·10-s + 1.24·11-s + 4.57·13-s − 3.45·14-s − 3.14·16-s − 5.83·17-s + 0.930·19-s − 0.430·20-s − 2.60·22-s + 23-s − 4.96·25-s − 9.56·26-s + 3.91·28-s − 29-s + 6.08·31-s + 8.07·32-s + 12.1·34-s − 0.301·35-s + 2.01·37-s − 1.94·38-s + 0.138·40-s − 5.93·41-s + ⋯
L(s)  = 1  − 1.47·2-s + 1.18·4-s − 0.0814·5-s + 0.625·7-s − 0.268·8-s + 0.120·10-s + 0.376·11-s + 1.27·13-s − 0.924·14-s − 0.785·16-s − 1.41·17-s + 0.213·19-s − 0.0962·20-s − 0.555·22-s + 0.208·23-s − 0.993·25-s − 1.87·26-s + 0.739·28-s − 0.185·29-s + 1.09·31-s + 1.42·32-s + 2.09·34-s − 0.0509·35-s + 0.331·37-s − 0.315·38-s + 0.0218·40-s − 0.926·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-1$
Analytic conductor: \(47.9341\)
Root analytic conductor: \(6.92345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + 2.08T + 2T^{2} \)
5 \( 1 + 0.182T + 5T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 4.57T + 13T^{2} \)
17 \( 1 + 5.83T + 17T^{2} \)
19 \( 1 - 0.930T + 19T^{2} \)
31 \( 1 - 6.08T + 31T^{2} \)
37 \( 1 - 2.01T + 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 - 6.30T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 0.787T + 53T^{2} \)
59 \( 1 + 0.995T + 59T^{2} \)
61 \( 1 + 8.77T + 61T^{2} \)
67 \( 1 - 0.890T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 0.935T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 - 7.42T + 89T^{2} \)
97 \( 1 - 0.195T + 97T^{2} \)
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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

−8.094607209093613879611223635751, −7.18888405236233803289809018043, −6.54503199742563935103218451752, −5.88459252836118794588632784349, −4.68703163180846966530287816313, −4.13614818848542521624874410827, −2.96864378348622951996328049665, −1.84724171333059420109396472169, −1.27164973612789331083023278094, 0, 1.27164973612789331083023278094, 1.84724171333059420109396472169, 2.96864378348622951996328049665, 4.13614818848542521624874410827, 4.68703163180846966530287816313, 5.88459252836118794588632784349, 6.54503199742563935103218451752, 7.18888405236233803289809018043, 8.094607209093613879611223635751

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