Properties

Degree $2$
Conductor $190$
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 − 7-s + 8-s − 2·9-s + 10-s + 12-s − 13-s − 14-s + 15-s + 16-s − 3·17-s − 2·18-s + 19-s + 20-s − 21-s + 3·23-s + 24-s + 25-s − 26-s − 5·27-s − 28-s − 3·29-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 − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.962·27-s − 0.188·28-s − 0.557·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190\)    =    \(2 \cdot 5 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{190} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 190,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.972462671\)
\(L(\frac12)\) \(\approx\) \(1.972462671\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.67798433560318, −19.12135773935416, −17.79391206371051, −17.02379969264462, −16.00772317824799, −15.08731769042180, −14.30858169472275, −13.56058406243046, −12.80305233753891, −11.70029699633934, −10.71176397114237, −9.514370507642223, −8.598780862883343, −7.285255719548759, −6.164052911988142, −5.042324572979757, −3.513840281870474, −2.337819964888564, 2.337819964888564, 3.513840281870474, 5.042324572979757, 6.164052911988142, 7.285255719548759, 8.598780862883343, 9.514370507642223, 10.71176397114237, 11.70029699633934, 12.80305233753891, 13.56058406243046, 14.30858169472275, 15.08731769042180, 16.00772317824799, 17.02379969264462, 17.79391206371051, 19.12135773935416, 19.67798433560318

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