Properties

Label 2-116886-1.1-c1-0-13
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 + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 12-s + 13-s + 14-s − 15-s + 16-s − 4·17-s − 18-s − 4·19-s + 20-s + 21-s + 23-s + 24-s − 4·25-s − 26-s − 27-s − 28-s + 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 + 1/3·9-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 − 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.182·30-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 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−13.65365656041717, −13.40540652015609, −12.80852873693235, −12.30270516062754, −11.82731915607687, −11.32242944797928, −10.88636084013231, −10.33324004444666, −9.987100362536118, −9.562959558399279, −8.904514856019011, −8.385327272772315, −8.176749457408279, −7.085206146092807, −6.972054368934472, −6.340110011701636, −5.953896753081534, −5.401823967300552, −4.656197704837293, −4.176147517427489, −3.481540812682204, −2.677195041773116, −2.168705237335655, −1.525697887894281, −0.7247359278412741, 0, 0.7247359278412741, 1.525697887894281, 2.168705237335655, 2.677195041773116, 3.481540812682204, 4.176147517427489, 4.656197704837293, 5.401823967300552, 5.953896753081534, 6.340110011701636, 6.972054368934472, 7.085206146092807, 8.176749457408279, 8.385327272772315, 8.904514856019011, 9.562959558399279, 9.987100362536118, 10.33324004444666, 10.88636084013231, 11.32242944797928, 11.82731915607687, 12.30270516062754, 12.80852873693235, 13.40540652015609, 13.65365656041717

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