Properties

Label 2-6018-1.1-c1-0-86
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 + 3.17·5-s + 6-s − 4.61·7-s − 8-s + 9-s − 3.17·10-s − 0.697·11-s − 12-s + 2.07·13-s + 4.61·14-s − 3.17·15-s + 16-s + 17-s − 18-s − 4.21·19-s + 3.17·20-s + 4.61·21-s + 0.697·22-s + 1.76·23-s + 24-s + 5.06·25-s − 2.07·26-s − 27-s − 4.61·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.41·5-s + 0.408·6-s − 1.74·7-s − 0.353·8-s + 0.333·9-s − 1.00·10-s − 0.210·11-s − 0.288·12-s + 0.575·13-s + 1.23·14-s − 0.819·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.967·19-s + 0.709·20-s + 1.00·21-s + 0.148·22-s + 0.368·23-s + 0.204·24-s + 1.01·25-s − 0.406·26-s − 0.192·27-s − 0.871·28-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 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + 0.697T + 11T^{2} \)
13 \( 1 - 2.07T + 13T^{2} \)
19 \( 1 + 4.21T + 19T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 - 6.98T + 37T^{2} \)
41 \( 1 + 9.67T + 41T^{2} \)
43 \( 1 + 0.221T + 43T^{2} \)
47 \( 1 - 4.09T + 47T^{2} \)
53 \( 1 + 4.12T + 53T^{2} \)
61 \( 1 - 3.42T + 61T^{2} \)
67 \( 1 + 9.96T + 67T^{2} \)
71 \( 1 - 3.78T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 - 3.39T + 79T^{2} \)
83 \( 1 - 3.65T + 83T^{2} \)
89 \( 1 + 3.75T + 89T^{2} \)
97 \( 1 - 7.20T + 97T^{2} \)
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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

−7.65791856517727707120503169966, −6.71252601677654233980778382305, −6.36212490836797642704348009551, −5.90902314951252538301697003726, −5.14467303882778625572383068487, −3.92155346225149546931876185181, −2.99184513728796674846617657926, −2.22944094666046863515652284800, −1.18134903739465789389831651384, 0, 1.18134903739465789389831651384, 2.22944094666046863515652284800, 2.99184513728796674846617657926, 3.92155346225149546931876185181, 5.14467303882778625572383068487, 5.90902314951252538301697003726, 6.36212490836797642704348009551, 6.71252601677654233980778382305, 7.65791856517727707120503169966

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