Properties

Label 2-116886-1.1-c1-0-46
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 + 6-s + 7-s − 8-s + 9-s − 12-s + 13-s − 14-s + 16-s − 5·17-s − 18-s + 5·19-s − 21-s + 23-s + 24-s − 5·25-s − 26-s − 27-s + 28-s − 9·29-s − 5·31-s − 32-s + 5·34-s + 36-s − 6·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 1.14·19-s − 0.218·21-s + 0.208·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 1.67·29-s − 0.898·31-s − 0.176·32-s + 0.857·34-s + 1/6·36-s − 0.986·37-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 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.02042931506352, −13.38965505142272, −13.24619761218577, −12.54502432374289, −11.84479456496137, −11.59644608561732, −11.18525722515065, −10.76735174158531, −10.12867892377833, −9.721974822089536, −9.218620425662227, −8.672836993648774, −8.242878449201115, −7.592114771639656, −7.062539607567995, −6.841577846629140, −5.994668097688107, −5.603183947656009, −5.068660082574814, −4.483413763506892, −3.656131825636400, −3.329133787534726, −2.337453073922629, −1.662251217178524, −1.381088906164290, 0, 0, 1.381088906164290, 1.662251217178524, 2.337453073922629, 3.329133787534726, 3.656131825636400, 4.483413763506892, 5.068660082574814, 5.603183947656009, 5.994668097688107, 6.841577846629140, 7.062539607567995, 7.592114771639656, 8.242878449201115, 8.672836993648774, 9.218620425662227, 9.721974822089536, 10.12867892377833, 10.76735174158531, 11.18525722515065, 11.59644608561732, 11.84479456496137, 12.54502432374289, 13.24619761218577, 13.38965505142272, 14.02042931506352

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