Properties

Label 2-116886-1.1-c1-0-22
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 − 4·5-s − 6-s − 7-s − 8-s + 9-s + 4·10-s + 12-s + 14-s − 4·15-s + 16-s + 2·17-s − 18-s − 4·20-s − 21-s − 23-s − 24-s + 11·25-s + 27-s − 28-s + 10·29-s + 4·30-s + 4·31-s − 32-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s + 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.894·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s − 0.188·28-s + 1.85·29-s + 0.730·30-s + 0.718·31-s − 0.176·32-s − 0.342·34-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.95589511700804, −13.28936828558120, −12.71385587945590, −12.17741382950846, −11.98983528982117, −11.40902774751553, −10.93682929425895, −10.28679875751849, −10.02980379540578, −9.326434426750139, −8.785416682302402, −8.318387549542352, −8.033884209074442, −7.538183934181614, −7.044303442831284, −6.573317098153713, −5.991154657387822, −5.074086517057534, −4.506251633211210, −3.998775113048083, −3.430492407258075, −2.888708895514839, −2.465986305906355, −1.321441962106959, −0.7958140478030702, 0, 0.7958140478030702, 1.321441962106959, 2.465986305906355, 2.888708895514839, 3.430492407258075, 3.998775113048083, 4.506251633211210, 5.074086517057534, 5.991154657387822, 6.573317098153713, 7.044303442831284, 7.538183934181614, 8.033884209074442, 8.318387549542352, 8.785416682302402, 9.326434426750139, 10.02980379540578, 10.28679875751849, 10.93682929425895, 11.40902774751553, 11.98983528982117, 12.17741382950846, 12.71385587945590, 13.28936828558120, 13.95589511700804

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