Properties

Label 2-86190-1.1-c1-0-5
Degree $2$
Conductor $86190$
Sign $1$
Analytic cond. $688.230$
Root an. cond. $26.2341$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·7-s − 8-s + 9-s − 10-s − 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 2·21-s + 4·23-s + 24-s + 25-s − 27-s − 2·28-s + 2·29-s + 30-s − 32-s + 34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.534·14-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 + 0.436·21-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.182·30-s − 0.176·32-s + 0.171·34-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: $1$
Analytic conductor: \(688.230\)
Root analytic conductor: \(26.2341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 86190,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.073961882\)
\(L(\frac12)\) \(\approx\) \(1.073961882\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.86941534409713, −13.24140294036984, −12.92094162071520, −12.27547365137673, −12.11055227631500, −11.09145304863419, −10.96657838696849, −10.48535180080175, −9.868156123670858, −9.470695607113220, −8.953986201156097, −8.564349424805281, −7.739230716054679, −7.316686853491154, −6.658917152097452, −6.344661294133432, −5.799459704352846, −5.293541179109108, −4.435848280728861, −4.047922323077936, −3.057696183163841, −2.633659917981445, −1.902890533553462, −1.101511876631257, −0.4342870498408680, 0.4342870498408680, 1.101511876631257, 1.902890533553462, 2.633659917981445, 3.057696183163841, 4.047922323077936, 4.435848280728861, 5.293541179109108, 5.799459704352846, 6.344661294133432, 6.658917152097452, 7.316686853491154, 7.739230716054679, 8.564349424805281, 8.953986201156097, 9.470695607113220, 9.868156123670858, 10.48535180080175, 10.96657838696849, 11.09145304863419, 12.11055227631500, 12.27547365137673, 12.92094162071520, 13.24140294036984, 13.86941534409713

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