Properties

Label 2-6003-1.1-c1-0-97
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.96·2-s + 1.86·4-s − 2.31·5-s + 5.13·7-s + 0.268·8-s + 4.54·10-s + 2.78·11-s + 1.47·13-s − 10.1·14-s − 4.25·16-s + 1.99·17-s + 2.47·19-s − 4.31·20-s − 5.47·22-s + 23-s + 0.352·25-s − 2.89·26-s + 9.57·28-s − 29-s + 5.51·31-s + 7.82·32-s − 3.92·34-s − 11.8·35-s + 7.95·37-s − 4.85·38-s − 0.620·40-s − 2.86·41-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.931·4-s − 1.03·5-s + 1.94·7-s + 0.0948·8-s + 1.43·10-s + 0.839·11-s + 0.408·13-s − 2.69·14-s − 1.06·16-s + 0.484·17-s + 0.567·19-s − 0.964·20-s − 1.16·22-s + 0.208·23-s + 0.0705·25-s − 0.567·26-s + 1.81·28-s − 0.185·29-s + 0.990·31-s + 1.38·32-s − 0.672·34-s − 2.00·35-s + 1.30·37-s − 0.788·38-s − 0.0981·40-s − 0.447·41-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: \(0\)
Selberg data: \((2,\ 6003,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.223090417\)
\(L(\frac12)\) \(\approx\) \(1.223090417\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.96T + 2T^{2} \)
5 \( 1 + 2.31T + 5T^{2} \)
7 \( 1 - 5.13T + 7T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 1.47T + 13T^{2} \)
17 \( 1 - 1.99T + 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
31 \( 1 - 5.51T + 31T^{2} \)
37 \( 1 - 7.95T + 37T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
43 \( 1 - 6.65T + 43T^{2} \)
47 \( 1 - 1.23T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 - 7.80T + 61T^{2} \)
67 \( 1 + 4.27T + 67T^{2} \)
71 \( 1 - 3.71T + 71T^{2} \)
73 \( 1 + 2.30T + 73T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 - 1.97T + 83T^{2} \)
89 \( 1 - 4.99T + 89T^{2} \)
97 \( 1 + 2.35T + 97T^{2} \)
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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

−8.107389200797217068592181518708, −7.66861369901240853041470089290, −7.17229405137029877783454709864, −6.13745309685545607993928101883, −5.08092236493279126971278576869, −4.40120798539928177902193290092, −3.77448005259922700904815169339, −2.41436001820516313211642507914, −1.37252798277468771351738588484, −0.849292558987414778674070010264, 0.849292558987414778674070010264, 1.37252798277468771351738588484, 2.41436001820516313211642507914, 3.77448005259922700904815169339, 4.40120798539928177902193290092, 5.08092236493279126971278576869, 6.13745309685545607993928101883, 7.17229405137029877783454709864, 7.66861369901240853041470089290, 8.107389200797217068592181518708

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