Properties

Label 2-116886-1.1-c1-0-4
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 $0$

Origins

Downloads

Learn more

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 − 4·19-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 + ⋯
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 − 0.917·19-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 + ⋯

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: \(0\)
Selberg data: \((2,\ 116886,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9981934612\)
\(L(\frac12)\) \(\approx\) \(0.9981934612\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.36794530112170, −12.99340586504185, −12.49085892327057, −12.02928899028395, −11.65707005453137, −10.96824927020827, −10.89815921427642, −10.12455293919369, −9.768503156139404, −9.238042734322900, −8.583898451744441, −8.150108500857149, −7.630914202747838, −7.288611313874080, −6.613185461100911, −6.105712341338618, −5.663969007930634, −5.063952626445933, −4.212890488433383, −3.801520055947894, −3.318705086857256, −2.492187904031475, −1.793028537005649, −0.9337319952227127, −0.4543487734814837, 0.4543487734814837, 0.9337319952227127, 1.793028537005649, 2.492187904031475, 3.318705086857256, 3.801520055947894, 4.212890488433383, 5.063952626445933, 5.663969007930634, 6.105712341338618, 6.613185461100911, 7.288611313874080, 7.630914202747838, 8.150108500857149, 8.583898451744441, 9.238042734322900, 9.768503156139404, 10.12455293919369, 10.89815921427642, 10.96824927020827, 11.65707005453137, 12.02928899028395, 12.49085892327057, 12.99340586504185, 13.36794530112170

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