Properties

Label 2-6018-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 3.58·5-s + 6-s − 2.07·7-s − 8-s + 9-s − 3.58·10-s − 4.55·11-s − 12-s − 5.49·13-s + 2.07·14-s − 3.58·15-s + 16-s + 17-s − 18-s + 4.37·19-s + 3.58·20-s + 2.07·21-s + 4.55·22-s + 7.83·23-s + 24-s + 7.86·25-s + 5.49·26-s − 27-s − 2.07·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s − 0.782·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s − 1.37·11-s − 0.288·12-s − 1.52·13-s + 0.553·14-s − 0.926·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 1.00·19-s + 0.802·20-s + 0.451·21-s + 0.971·22-s + 1.63·23-s + 0.204·24-s + 1.57·25-s + 1.07·26-s − 0.192·27-s − 0.391·28-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: \(0\)
Selberg data: \((2,\ 6018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.035674353\)
\(L(\frac12)\) \(\approx\) \(1.035674353\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.58T + 5T^{2} \)
7 \( 1 + 2.07T + 7T^{2} \)
11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 + 5.49T + 13T^{2} \)
19 \( 1 - 4.37T + 19T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + 2.79T + 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + 0.724T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 0.730T + 47T^{2} \)
53 \( 1 - 1.71T + 53T^{2} \)
61 \( 1 + 1.71T + 61T^{2} \)
67 \( 1 + 7.85T + 67T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 5.20T + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + 6.72T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.007490395224570252801016395310, −7.11639568239354579500636942077, −6.91700758323426861458482611492, −5.84144123711035923075304139974, −5.34122261627283271406154689110, −4.94211419556030341128778865563, −3.24333124789373198610313730947, −2.60324552372394777967143358872, −1.82046215184681348639973338087, −0.59307268374123515490891182617, 0.59307268374123515490891182617, 1.82046215184681348639973338087, 2.60324552372394777967143358872, 3.24333124789373198610313730947, 4.94211419556030341128778865563, 5.34122261627283271406154689110, 5.84144123711035923075304139974, 6.91700758323426861458482611492, 7.11639568239354579500636942077, 8.007490395224570252801016395310

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