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 1

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 − 4·23-s + 24-s + 25-s − 27-s + 4·28-s + 2·29-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 + 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 − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.755·28-s + 0.371·29-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  =  1
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.26336256860447, −13.76892713713601, −13.22605164662550, −12.35194810652821, −12.07897654403653, −11.54862225084596, −11.20926658801079, −10.82762689844465, −10.05847602468756, −9.703511887023988, −9.275177329734965, −8.564245817019789, −8.105565362112772, −7.717737852455716, −7.007275469925795, −6.556909874844540, −6.004026334005883, −5.375602869633906, −4.944389349194350, −4.307274450267029, −3.650296919867765, −2.879437798483499, −1.939119384210061, −1.481889983724610, −1.123262045511335, 0, 1.123262045511335, 1.481889983724610, 1.939119384210061, 2.879437798483499, 3.650296919867765, 4.307274450267029, 4.944389349194350, 5.375602869633906, 6.004026334005883, 6.556909874844540, 7.007275469925795, 7.717737852455716, 8.105565362112772, 8.564245817019789, 9.275177329734965, 9.703511887023988, 10.05847602468756, 10.82762689844465, 11.20926658801079, 11.54862225084596, 12.07897654403653, 12.35194810652821, 13.22605164662550, 13.76892713713601, 14.26336256860447

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