Properties

Label 2-6018-1.1-c1-0-94
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 − 1.98·5-s − 6-s − 2.53·7-s − 8-s + 9-s + 1.98·10-s + 6.11·11-s + 12-s − 1.71·13-s + 2.53·14-s − 1.98·15-s + 16-s + 17-s − 18-s + 4.16·19-s − 1.98·20-s − 2.53·21-s − 6.11·22-s − 8.88·23-s − 24-s − 1.07·25-s + 1.71·26-s + 27-s − 2.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.886·5-s − 0.408·6-s − 0.958·7-s − 0.353·8-s + 0.333·9-s + 0.626·10-s + 1.84·11-s + 0.288·12-s − 0.475·13-s + 0.677·14-s − 0.511·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.956·19-s − 0.443·20-s − 0.553·21-s − 1.30·22-s − 1.85·23-s − 0.204·24-s − 0.214·25-s + 0.336·26-s + 0.192·27-s − 0.479·28-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 \( 1 + 1.98T + 5T^{2} \)
7 \( 1 + 2.53T + 7T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
13 \( 1 + 1.71T + 13T^{2} \)
19 \( 1 - 4.16T + 19T^{2} \)
23 \( 1 + 8.88T + 23T^{2} \)
29 \( 1 - 2.92T + 29T^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 - 0.684T + 37T^{2} \)
41 \( 1 - 6.28T + 41T^{2} \)
43 \( 1 - 0.490T + 43T^{2} \)
47 \( 1 + 2.59T + 47T^{2} \)
53 \( 1 + 8.69T + 53T^{2} \)
61 \( 1 - 2.45T + 61T^{2} \)
67 \( 1 + 2.32T + 67T^{2} \)
71 \( 1 + 6.70T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 3.42T + 79T^{2} \)
83 \( 1 + 0.548T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + 4.28T + 97T^{2} \)
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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

−7.67011260166335765053084666141, −7.29603573546476119904430439413, −6.43682138768638689445677163660, −5.93539043906562017959425969685, −4.57808301919095337019137604943, −3.67699880365849438175712318401, −3.44309902658862931492857642597, −2.24844116542332086765954966305, −1.22520379055880673771344961768, 0, 1.22520379055880673771344961768, 2.24844116542332086765954966305, 3.44309902658862931492857642597, 3.67699880365849438175712318401, 4.57808301919095337019137604943, 5.93539043906562017959425969685, 6.43682138768638689445677163660, 7.29603573546476119904430439413, 7.67011260166335765053084666141

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