Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17 $
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 − 5-s + 6-s − 4·7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 4·14-s + 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s + 4·21-s + 4·22-s + 24-s + 25-s − 27-s − 4·28-s + 6·29-s − 30-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.872·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 1.11·29-s − 0.182·30-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{86190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7838512473$
$L(\frac12)$  $\approx$  $0.7838512473$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.89296640971396, −13.10883489528942, −12.85506216275974, −12.46366765131574, −11.96342661636235, −11.34604353263848, −10.78363063939384, −10.38359372710725, −10.03372511098039, −9.531614802414662, −8.863541379763597, −8.439599707754638, −7.799784012762192, −7.293252091422171, −6.802785236099269, −6.318740060305386, −5.749350006340354, −5.317970401373011, −4.391934360589476, −3.969193406629892, −3.110426066743131, −2.679975967689770, −2.059135791172861, −0.7922575432106015, −0.4716763366496180, 0.4716763366496180, 0.7922575432106015, 2.059135791172861, 2.679975967689770, 3.110426066743131, 3.969193406629892, 4.391934360589476, 5.317970401373011, 5.749350006340354, 6.318740060305386, 6.802785236099269, 7.293252091422171, 7.799784012762192, 8.439599707754638, 8.863541379763597, 9.531614802414662, 10.03372511098039, 10.38359372710725, 10.78363063939384, 11.34604353263848, 11.96342661636235, 12.46366765131574, 12.85506216275974, 13.10883489528942, 13.89296640971396

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