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 + 2·7-s − 8-s + 9-s − 10-s − 5·11-s − 12-s − 2·14-s − 15-s + 16-s − 17-s − 18-s + 6·19-s + 20-s − 2·21-s + 5·22-s − 23-s + 24-s + 25-s − 27-s + 2·28-s − 7·29-s + 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.436·21-s + 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 1.29·29-s + 0.182·30-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  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.301268186$
$L(\frac12)$  $\approx$  $1.301268186$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.96132923257178, −13.35900496825412, −12.87415590281468, −12.41006327319394, −11.78535457876792, −11.17222531971960, −11.06013217329341, −10.47051447521229, −9.832360050589021, −9.584499616623956, −9.004035016329199, −8.177702625403662, −7.785184845573643, −7.593629025075450, −6.790207078126431, −6.191325511823012, −5.687870801723373, −5.046861738212736, −4.884147469819694, −3.924558426182474, −3.113342413816665, −2.495568707820919, −1.903052786794413, −1.184865084051091, −0.4580792914693690, 0.4580792914693690, 1.184865084051091, 1.903052786794413, 2.495568707820919, 3.113342413816665, 3.924558426182474, 4.884147469819694, 5.046861738212736, 5.687870801723373, 6.191325511823012, 6.790207078126431, 7.593629025075450, 7.785184845573643, 8.177702625403662, 9.004035016329199, 9.584499616623956, 9.832360050589021, 10.47051447521229, 11.06013217329341, 11.17222531971960, 11.78535457876792, 12.41006327319394, 12.87415590281468, 13.35900496825412, 13.96132923257178

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