Properties

Degree $2$
Conductor $86190$
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 − 7-s − 8-s + 9-s − 10-s + 5·11-s − 12-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 21-s − 5·22-s + 24-s + 25-s − 27-s − 28-s − 2·29-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 − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 1.06·22-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-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

Degree: \(2\)
Conductor: \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{86190} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 86190,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.11307217039460, −13.76020359492912, −13.02063438189027, −12.62692246269587, −11.99412704229928, −11.71172248063555, −11.17095041543187, −10.66374137091748, −9.944578191014634, −9.858136874358277, −9.187162153441035, −8.714639298533925, −8.276961037476030, −7.444608731245985, −6.959858690384173, −6.466723176164044, −6.216902335381883, −5.421936616009443, −4.997362073648646, −4.109511428261507, −3.612966129184623, −2.991280777640606, −2.011402131065928, −1.579222070697351, −0.8709720119403842, 0, 0.8709720119403842, 1.579222070697351, 2.011402131065928, 2.991280777640606, 3.612966129184623, 4.109511428261507, 4.997362073648646, 5.421936616009443, 6.216902335381883, 6.466723176164044, 6.959858690384173, 7.444608731245985, 8.276961037476030, 8.714639298533925, 9.187162153441035, 9.858136874358277, 9.944578191014634, 10.66374137091748, 11.17095041543187, 11.71172248063555, 11.99412704229928, 12.62692246269587, 13.02063438189027, 13.76020359492912, 14.11307217039460

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