Properties

Label 2-116886-1.1-c1-0-41
Degree $2$
Conductor $116886$
Sign $1$
Analytic cond. $933.339$
Root an. cond. $30.5506$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 6·13-s − 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 21-s − 23-s − 24-s − 25-s + 6·26-s + 27-s + 28-s − 10·29-s + 2·30-s + 5·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s − 1.85·29-s + 0.365·30-s + 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(116886\)    =    \(2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(933.339\)
Root analytic conductor: \(30.5506\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{116886} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 116886,\ (\ :1/2),\ 1)\)

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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−14.29621573899034, −13.52291197257566, −12.97637598063671, −12.75968235214118, −11.83494498326411, −11.61194653723012, −11.35127677929268, −10.60323209781604, −9.988565729299646, −9.754632638657298, −9.074608531862449, −8.683688143612089, −8.116005970484308, −7.714618211111949, −7.316339199722053, −6.834496414463078, −6.299482462611724, −5.384224990783863, −4.926762969335166, −4.210427788300279, −3.907050755915078, −3.060704624498286, −2.410289287248410, −2.062624628944817, −1.250888927739124, 0, 0, 1.250888927739124, 2.062624628944817, 2.410289287248410, 3.060704624498286, 3.907050755915078, 4.210427788300279, 4.926762969335166, 5.384224990783863, 6.299482462611724, 6.834496414463078, 7.316339199722053, 7.714618211111949, 8.116005970484308, 8.683688143612089, 9.074608531862449, 9.754632638657298, 9.988565729299646, 10.60323209781604, 11.35127677929268, 11.61194653723012, 11.83494498326411, 12.75968235214118, 12.97637598063671, 13.52291197257566, 14.29621573899034

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