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 + 1.96·5-s − 6-s − 3.61·7-s − 8-s + 9-s − 1.96·10-s + 0.322·11-s + 12-s + 2.69·13-s + 3.61·14-s + 1.96·15-s + 16-s − 17-s − 18-s − 3.90·19-s + 1.96·20-s − 3.61·21-s − 0.322·22-s − 4.79·23-s − 24-s − 1.12·25-s − 2.69·26-s + 27-s − 3.61·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.879·5-s − 0.408·6-s − 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.622·10-s + 0.0971·11-s + 0.288·12-s + 0.748·13-s + 0.966·14-s + 0.507·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.896·19-s + 0.439·20-s − 0.789·21-s − 0.0686·22-s − 0.999·23-s − 0.204·24-s − 0.225·25-s − 0.528·26-s + 0.192·27-s − 0.683·28-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  =  $(2,\ 6018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.656425489$
$L(\frac12)$  $\approx$  $1.656425489$
$L(\frac{3}{2})$   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 \( 1 - 1.96T + 5T^{2} \)
7 \( 1 + 3.61T + 7T^{2} \)
11 \( 1 - 0.322T + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
19 \( 1 + 3.90T + 19T^{2} \)
23 \( 1 + 4.79T + 23T^{2} \)
29 \( 1 + 0.708T + 29T^{2} \)
31 \( 1 - 6.38T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 - 1.55T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 1.01T + 53T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 - 7.07T + 71T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 + 0.0479T + 79T^{2} \)
83 \( 1 - 0.102T + 83T^{2} \)
89 \( 1 - 3.15T + 89T^{2} \)
97 \( 1 - 2.11T + 97T^{2} \)
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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

−8.300354090189132048580733820081, −7.39717643354714115787234663929, −6.58171421742050750902427484028, −6.21528970363499698465289635524, −5.52658004436616627410384149511, −4.15312137529228180420744009098, −3.53509343352594899718176506426, −2.52315163885319717623102513985, −1.99400511970926957029571025385, −0.71196915385092737824283181565, 0.71196915385092737824283181565, 1.99400511970926957029571025385, 2.52315163885319717623102513985, 3.53509343352594899718176506426, 4.15312137529228180420744009098, 5.52658004436616627410384149511, 6.21528970363499698465289635524, 6.58171421742050750902427484028, 7.39717643354714115787234663929, 8.300354090189132048580733820081

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