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 + 3.05·5-s − 6-s + 1.53·7-s − 8-s + 9-s − 3.05·10-s − 3.38·11-s + 12-s − 1.02·13-s − 1.53·14-s + 3.05·15-s + 16-s − 17-s − 18-s + 3.86·19-s + 3.05·20-s + 1.53·21-s + 3.38·22-s + 3.31·23-s − 24-s + 4.34·25-s + 1.02·26-s + 27-s + 1.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.36·5-s − 0.408·6-s + 0.580·7-s − 0.353·8-s + 0.333·9-s − 0.966·10-s − 1.01·11-s + 0.288·12-s − 0.283·13-s − 0.410·14-s + 0.789·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.885·19-s + 0.683·20-s + 0.335·21-s + 0.721·22-s + 0.691·23-s − 0.204·24-s + 0.869·25-s + 0.200·26-s + 0.192·27-s + 0.290·28-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  =  $(2,\ 6018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.582476612$
$L(\frac12)$  $\approx$  $2.582476612$
$L(\frac{3}{2})$   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 \( 1 - 3.05T + 5T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
19 \( 1 - 3.86T + 19T^{2} \)
23 \( 1 - 3.31T + 23T^{2} \)
29 \( 1 - 9.99T + 29T^{2} \)
31 \( 1 + 0.616T + 31T^{2} \)
37 \( 1 - 6.96T + 37T^{2} \)
41 \( 1 - 3.75T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 + 4.62T + 53T^{2} \)
61 \( 1 + 7.29T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 - 3.33T + 71T^{2} \)
73 \( 1 + 6.75T + 73T^{2} \)
79 \( 1 - 9.75T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 6.45T + 89T^{2} \)
97 \( 1 - 3.62T + 97T^{2} \)
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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

−8.108680768078744156161699776754, −7.60209715949024757195496855614, −6.72179416306849607011098980578, −6.09146952357423584217797024050, −5.14452013025369148099512385013, −4.73272109022745040524684943752, −3.21893926815618757636543110178, −2.57684737901484364137224665864, −1.88422329340514350656408090316, −0.946402405965776991617956002487, 0.946402405965776991617956002487, 1.88422329340514350656408090316, 2.57684737901484364137224665864, 3.21893926815618757636543110178, 4.73272109022745040524684943752, 5.14452013025369148099512385013, 6.09146952357423584217797024050, 6.72179416306849607011098980578, 7.60209715949024757195496855614, 8.108680768078744156161699776754

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