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 − 1.37·5-s − 6-s + 2.44·7-s − 8-s + 9-s + 1.37·10-s + 4.53·11-s + 12-s − 3.92·13-s − 2.44·14-s − 1.37·15-s + 16-s − 17-s − 18-s + 1.28·19-s − 1.37·20-s + 2.44·21-s − 4.53·22-s + 3.17·23-s − 24-s − 3.11·25-s + 3.92·26-s + 27-s + 2.44·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.613·5-s − 0.408·6-s + 0.924·7-s − 0.353·8-s + 0.333·9-s + 0.434·10-s + 1.36·11-s + 0.288·12-s − 1.08·13-s − 0.653·14-s − 0.354·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.294·19-s − 0.306·20-s + 0.533·21-s − 0.967·22-s + 0.662·23-s − 0.204·24-s − 0.623·25-s + 0.769·26-s + 0.192·27-s + 0.462·28-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  =  $(2,\ 6018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.863481531$
$L(\frac12)$  $\approx$  $1.863481531$
$L(\frac{3}{2})$   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 \( 1 + 1.37T + 5T^{2} \)
7 \( 1 - 2.44T + 7T^{2} \)
11 \( 1 - 4.53T + 11T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
29 \( 1 + 4.44T + 29T^{2} \)
31 \( 1 - 5.90T + 31T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 - 5.71T + 41T^{2} \)
43 \( 1 + 3.32T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 6.22T + 53T^{2} \)
61 \( 1 - 4.28T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + 6.25T + 97T^{2} \)
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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

−8.003372396709542343923093929595, −7.54261122616532281262345990960, −7.00676935436736443874009134267, −6.14491560834587093909395635820, −5.09463272157498052389589033493, −4.31772639145234270503273096179, −3.65005391764941215143034594835, −2.59477830679039228778397139738, −1.78325398793599806838254188805, −0.801537594444080251525370606947, 0.801537594444080251525370606947, 1.78325398793599806838254188805, 2.59477830679039228778397139738, 3.65005391764941215143034594835, 4.31772639145234270503273096179, 5.09463272157498052389589033493, 6.14491560834587093909395635820, 7.00676935436736443874009134267, 7.54261122616532281262345990960, 8.003372396709542343923093929595

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