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 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s + 10-s − 5·11-s − 12-s − 3·14-s − 15-s + 16-s + 17-s + 18-s − 2·19-s + 20-s + 3·21-s − 5·22-s − 23-s − 24-s + 25-s − 27-s − 3·28-s − 6·29-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 − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.50·11-s − 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s + 0.654·21-s − 1.06·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.566·28-s − 1.11·29-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  =  2
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−14.34240839216218, −13.57829738512970, −13.44426258286303, −12.92421695790359, −12.58452436348888, −12.10734636677093, −11.47973008700149, −10.92218330731067, −10.47200616741441, −9.987535930746728, −9.718471140114854, −8.937503537127608, −8.264053164507094, −7.749286463845670, −7.011435284229597, −6.729850638587490, −6.076850551513667, −5.642984964680813, −5.147305718222829, −4.726335415660347, −3.867960418839571, −3.255870101722723, −2.885318260506813, −2.008539112818131, −1.504533938889334, 0, 0, 1.504533938889334, 2.008539112818131, 2.885318260506813, 3.255870101722723, 3.867960418839571, 4.726335415660347, 5.147305718222829, 5.642984964680813, 6.076850551513667, 6.729850638587490, 7.011435284229597, 7.749286463845670, 8.264053164507094, 8.937503537127608, 9.718471140114854, 9.987535930746728, 10.47200616741441, 10.92218330731067, 11.47973008700149, 12.10734636677093, 12.58452436348888, 12.92421695790359, 13.44426258286303, 13.57829738512970, 14.34240839216218

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