Properties

Label 2-6018-1.1-c1-0-87
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 + 4.32·5-s + 6-s + 3.19·7-s − 8-s + 9-s − 4.32·10-s + 4.37·11-s − 12-s + 0.617·13-s − 3.19·14-s − 4.32·15-s + 16-s + 17-s − 18-s + 1.19·19-s + 4.32·20-s − 3.19·21-s − 4.37·22-s + 4.91·23-s + 24-s + 13.7·25-s − 0.617·26-s − 27-s + 3.19·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.93·5-s + 0.408·6-s + 1.20·7-s − 0.353·8-s + 0.333·9-s − 1.36·10-s + 1.32·11-s − 0.288·12-s + 0.171·13-s − 0.853·14-s − 1.11·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.273·19-s + 0.967·20-s − 0.696·21-s − 0.933·22-s + 1.02·23-s + 0.204·24-s + 2.74·25-s − 0.121·26-s − 0.192·27-s + 0.603·28-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

\(L(1)\) \(\approx\) \(2.579172252\)
\(L(\frac12)\) \(\approx\) \(2.579172252\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.32T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 - 0.617T + 13T^{2} \)
19 \( 1 - 1.19T + 19T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 - 5.98T + 29T^{2} \)
31 \( 1 + 1.28T + 31T^{2} \)
37 \( 1 + 6.55T + 37T^{2} \)
41 \( 1 + 7.41T + 41T^{2} \)
43 \( 1 + 4.76T + 43T^{2} \)
47 \( 1 - 2.37T + 47T^{2} \)
53 \( 1 + 3.41T + 53T^{2} \)
61 \( 1 + 6.96T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 + 0.350T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 0.864T + 79T^{2} \)
83 \( 1 + 2.76T + 83T^{2} \)
89 \( 1 - 3.17T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
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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

−8.292563655713248723734518446678, −7.20192638981982567456402256904, −6.56586744280538494867884978022, −6.15409162586186224539587620180, −5.10359264920168518554914100964, −4.97176198166899851016518602749, −3.52292256263032488143216420811, −2.37080920451915986394013477281, −1.47155375206080000266496355911, −1.18835383015071006714047264235, 1.18835383015071006714047264235, 1.47155375206080000266496355911, 2.37080920451915986394013477281, 3.52292256263032488143216420811, 4.97176198166899851016518602749, 5.10359264920168518554914100964, 6.15409162586186224539587620180, 6.56586744280538494867884978022, 7.20192638981982567456402256904, 8.292563655713248723734518446678

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