Properties

Label 2-116886-1.1-c1-0-6
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 − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 12-s − 5·13-s + 14-s − 3·15-s + 16-s − 18-s − 8·19-s − 3·20-s − 21-s − 23-s − 24-s + 4·25-s + 5·26-s + 27-s − 28-s − 3·29-s + 3·30-s + 2·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.288·12-s − 1.38·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.670·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.980·26-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.547·30-s + 0.359·31-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: Trivial
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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.90124258899132, −13.32568028594955, −12.59045765366392, −12.33442900463657, −12.04882401556499, −11.34640903025292, −10.81016114569005, −10.45437507202053, −9.867226215776274, −9.372563043797246, −8.868895767321899, −8.391271527079820, −7.852606106274363, −7.639456356837657, −6.987571851279780, −6.553626482494505, −6.040574065617119, −4.926700058020733, −4.712337912280693, −3.971590151877353, −3.382692237120581, −2.997445924392166, −2.075166742172502, −1.807158692828441, −0.5076440639193479, 0, 0.5076440639193479, 1.807158692828441, 2.075166742172502, 2.997445924392166, 3.382692237120581, 3.971590151877353, 4.712337912280693, 4.926700058020733, 6.040574065617119, 6.553626482494505, 6.987571851279780, 7.639456356837657, 7.852606106274363, 8.391271527079820, 8.868895767321899, 9.372563043797246, 9.867226215776274, 10.45437507202053, 10.81016114569005, 11.34640903025292, 12.04882401556499, 12.33442900463657, 12.59045765366392, 13.32568028594955, 13.90124258899132

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