Properties

Degree $2$
Conductor $86190$
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 − 8·19-s − 20-s − 4·21-s − 4·22-s + 24-s + 25-s − 27-s + 4·28-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 + 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 − 1.83·19-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 − 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: \(0\)
Selberg data: \((2,\ 86190,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709634547\)
\(L(\frac12)\) \(\approx\) \(1.709634547\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.14091106753198, −13.22282207453853, −12.93327006115624, −12.04038074041184, −11.88309582195052, −11.33025573436938, −11.09883885815054, −10.50147144662579, −9.995369258459001, −9.416081942273846, −8.722768476032131, −8.294823799778245, −8.103444895730683, −7.284017445936533, −6.789182963093747, −6.409662666931227, −5.698443043499950, −5.068477763081494, −4.523429030195154, −3.988121687564624, −3.463275093354634, −2.232715866156672, −1.944015281312604, −1.146756243869350, −0.5520747855736981, 0.5520747855736981, 1.146756243869350, 1.944015281312604, 2.232715866156672, 3.463275093354634, 3.988121687564624, 4.523429030195154, 5.068477763081494, 5.698443043499950, 6.409662666931227, 6.789182963093747, 7.284017445936533, 8.103444895730683, 8.294823799778245, 8.722768476032131, 9.416081942273846, 9.995369258459001, 10.50147144662579, 11.09883885815054, 11.33025573436938, 11.88309582195052, 12.04038074041184, 12.93327006115624, 13.22282207453853, 14.14091106753198

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