Properties

Label 2-6018-1.1-c1-0-78
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 + 3.88·5-s + 6-s − 1.31·7-s + 8-s + 9-s + 3.88·10-s − 0.382·11-s + 12-s − 5.18·13-s − 1.31·14-s + 3.88·15-s + 16-s − 17-s + 18-s + 7.41·19-s + 3.88·20-s − 1.31·21-s − 0.382·22-s + 5.18·23-s + 24-s + 10.0·25-s − 5.18·26-s + 27-s − 1.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.73·5-s + 0.408·6-s − 0.496·7-s + 0.353·8-s + 0.333·9-s + 1.22·10-s − 0.115·11-s + 0.288·12-s − 1.43·13-s − 0.351·14-s + 1.00·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 1.70·19-s + 0.868·20-s − 0.286·21-s − 0.0816·22-s + 1.08·23-s + 0.204·24-s + 2.01·25-s − 1.01·26-s + 0.192·27-s − 0.248·28-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

\(L(1)\) \(\approx\) \(5.461190299\)
\(L(\frac12)\) \(\approx\) \(5.461190299\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.88T + 5T^{2} \)
7 \( 1 + 1.31T + 7T^{2} \)
11 \( 1 + 0.382T + 11T^{2} \)
13 \( 1 + 5.18T + 13T^{2} \)
19 \( 1 - 7.41T + 19T^{2} \)
23 \( 1 - 5.18T + 23T^{2} \)
29 \( 1 - 2.99T + 29T^{2} \)
31 \( 1 + 5.17T + 31T^{2} \)
37 \( 1 + 6.75T + 37T^{2} \)
41 \( 1 - 8.40T + 41T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 - 6.28T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 8.74T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−7.919652746395077647511569818935, −7.07975565342563770166924892426, −6.76930864238178640323921585045, −5.69518045789920359267134288149, −5.32027775279014452127418859444, −4.62437839905934813611765690375, −3.42047921699697407571735788649, −2.70970057917732146657526204829, −2.21297837280853149736232207829, −1.13818569483158793239177409395, 1.13818569483158793239177409395, 2.21297837280853149736232207829, 2.70970057917732146657526204829, 3.42047921699697407571735788649, 4.62437839905934813611765690375, 5.32027775279014452127418859444, 5.69518045789920359267134288149, 6.76930864238178640323921585045, 7.07975565342563770166924892426, 7.919652746395077647511569818935

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