Properties

Label 2-116886-1.1-c1-0-18
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 $1$

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 − 2·13-s − 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 2·20-s + 21-s + 23-s − 24-s − 25-s + 2·26-s + 27-s + 28-s + 2·29-s + 2·30-s + 4·31-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 − 0.554·13-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 + 0.392·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + 0.718·31-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.81209434081805, −13.25873768337327, −12.98084301343468, −12.24759328744777, −11.70509437698442, −11.44335390182668, −11.01650881175464, −10.25277265570293, −9.977408120145281, −9.341990897231104, −8.842353841106894, −8.311797069011382, −8.119098650459218, −7.402094906355796, −7.145670516051474, −6.445869143784325, −6.036856558623885, −4.945098666407316, −4.704593248960706, −4.052289625490557, −3.450167916347845, −2.773015813406504, −2.249138921346508, −1.627697903424404, −0.7465298466096718, 0, 0.7465298466096718, 1.627697903424404, 2.249138921346508, 2.773015813406504, 3.450167916347845, 4.052289625490557, 4.704593248960706, 4.945098666407316, 6.036856558623885, 6.445869143784325, 7.145670516051474, 7.402094906355796, 8.119098650459218, 8.311797069011382, 8.842353841106894, 9.341990897231104, 9.977408120145281, 10.25277265570293, 11.01650881175464, 11.44335390182668, 11.70509437698442, 12.24759328744777, 12.98084301343468, 13.25873768337327, 13.81209434081805

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