Properties

Label 2-116886-1.1-c1-0-12
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 − 6-s − 7-s − 8-s + 9-s + 12-s − 6·13-s + 14-s + 16-s − 4·17-s − 18-s + 2·19-s − 21-s − 23-s − 24-s − 5·25-s + 6·26-s + 27-s − 28-s − 2·29-s − 2·31-s − 32-s + 4·34-s + 36-s + 2·37-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 − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 1/6·36-s + 0.328·37-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: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.70702594182184, −13.45068925195889, −12.82291875973380, −12.39028910344360, −11.78386837990308, −11.50452132595799, −10.81384421435364, −10.15781184809919, −9.943626162003579, −9.289609631425366, −9.177231201310064, −8.391037060512817, −7.900414455683642, −7.515334869719742, −6.884107392357079, −6.661008952349902, −5.804582052639389, −5.264230385654246, −4.669220222143106, −3.946053569963770, −3.472339919381334, −2.632113587503121, −2.281437598258824, −1.757473614198894, −0.7078212811908110, 0, 0.7078212811908110, 1.757473614198894, 2.281437598258824, 2.632113587503121, 3.472339919381334, 3.946053569963770, 4.669220222143106, 5.264230385654246, 5.804582052639389, 6.661008952349902, 6.884107392357079, 7.515334869719742, 7.900414455683642, 8.391037060512817, 9.177231201310064, 9.289609631425366, 9.943626162003579, 10.15781184809919, 10.81384421435364, 11.50452132595799, 11.78386837990308, 12.39028910344360, 12.82291875973380, 13.45068925195889, 13.70702594182184

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