Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 17 \cdot 59 $
Sign $1$
Motivic weight 1
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 − 1.08·5-s − 6-s − 1.24·7-s − 8-s + 9-s + 1.08·10-s − 1.58·11-s + 12-s − 3.49·13-s + 1.24·14-s − 1.08·15-s + 16-s − 17-s − 18-s + 3.42·19-s − 1.08·20-s − 1.24·21-s + 1.58·22-s + 1.76·23-s − 24-s − 3.81·25-s + 3.49·26-s + 27-s − 1.24·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.486·5-s − 0.408·6-s − 0.471·7-s − 0.353·8-s + 0.333·9-s + 0.343·10-s − 0.478·11-s + 0.288·12-s − 0.970·13-s + 0.333·14-s − 0.280·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.786·19-s − 0.243·20-s − 0.272·21-s + 0.338·22-s + 0.368·23-s − 0.204·24-s − 0.763·25-s + 0.686·26-s + 0.192·27-s − 0.235·28-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

\( d \)  =  \(2\)
\( N \)  =  \(6018\)    =    \(2 \cdot 3 \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.090353495$
$L(\frac12)$  $\approx$  $1.090353495$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
17 \( 1 + T \)
59 \( 1 + T \)
good5 \( 1 + 1.08T + 5T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + 1.58T + 11T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 - 1.05T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + 0.263T + 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 + 7.04T + 47T^{2} \)
53 \( 1 - 8.92T + 53T^{2} \)
61 \( 1 - 2.97T + 61T^{2} \)
67 \( 1 - 1.69T + 67T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + 2.04T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 7.00T + 83T^{2} \)
89 \( 1 - 2.56T + 89T^{2} \)
97 \( 1 - 5.85T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.110999591390413694144530937996, −7.41819880358056582134014001045, −7.02337591055591966672985291895, −6.12527357508608537223664896786, −5.17490834166728502298569631407, −4.39390805266223494472711229883, −3.35200576990292072830294597469, −2.79286037934890520554847767591, −1.87683752195638375851255565029, −0.57390895122571145577143980477, 0.57390895122571145577143980477, 1.87683752195638375851255565029, 2.79286037934890520554847767591, 3.35200576990292072830294597469, 4.39390805266223494472711229883, 5.17490834166728502298569631407, 6.12527357508608537223664896786, 7.02337591055591966672985291895, 7.41819880358056582134014001045, 8.110999591390413694144530937996

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