Properties

Label 2-6018-1.1-c1-0-148
Degree $2$
Conductor $6018$
Sign $-1$
Analytic cond. $48.0539$
Root an. cond. $6.93209$
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-s + 3-s + 4-s + 1.83·5-s + 6-s − 4.13·7-s + 8-s + 9-s + 1.83·10-s − 4.28·11-s + 12-s + 0.501·13-s − 4.13·14-s + 1.83·15-s + 16-s − 17-s + 18-s + 4.55·19-s + 1.83·20-s − 4.13·21-s − 4.28·22-s − 6.84·23-s + 24-s − 1.64·25-s + 0.501·26-s + 27-s − 4.13·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.819·5-s + 0.408·6-s − 1.56·7-s + 0.353·8-s + 0.333·9-s + 0.579·10-s − 1.29·11-s + 0.288·12-s + 0.138·13-s − 1.10·14-s + 0.473·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.04·19-s + 0.409·20-s − 0.902·21-s − 0.913·22-s − 1.42·23-s + 0.204·24-s − 0.328·25-s + 0.0982·26-s + 0.192·27-s − 0.781·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6018\)    =    \(2 \cdot 3 \cdot 17 \cdot 59\)
Sign: $-1$
Analytic conductor: \(48.0539\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6018,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 - T \)
17 \( 1 + T \)
59 \( 1 + T \)
good5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 + 4.13T + 7T^{2} \)
11 \( 1 + 4.28T + 11T^{2} \)
13 \( 1 - 0.501T + 13T^{2} \)
19 \( 1 - 4.55T + 19T^{2} \)
23 \( 1 + 6.84T + 23T^{2} \)
29 \( 1 + 3.33T + 29T^{2} \)
31 \( 1 - 2.55T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 5.78T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
61 \( 1 - 3.98T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 6.04T + 71T^{2} \)
73 \( 1 + 6.23T + 73T^{2} \)
79 \( 1 - 9.82T + 79T^{2} \)
83 \( 1 - 5.35T + 83T^{2} \)
89 \( 1 + 5.53T + 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−7.64460156576727924367282549886, −6.89938625441091405173759445679, −6.16414247688147490920724427557, −5.67485157313699253595252550209, −4.91035751408397628720432928755, −3.82563453354281331858838946155, −3.19931978158229223853339151015, −2.55628191986725704637445220500, −1.72812162739170178230731661288, 0, 1.72812162739170178230731661288, 2.55628191986725704637445220500, 3.19931978158229223853339151015, 3.82563453354281331858838946155, 4.91035751408397628720432928755, 5.67485157313699253595252550209, 6.16414247688147490920724427557, 6.89938625441091405173759445679, 7.64460156576727924367282549886

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