Properties

Label 2-116886-1.1-c1-0-3
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

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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 − 2·17-s − 18-s + 2·20-s − 21-s + 23-s − 24-s − 25-s + 2·26-s + 27-s − 28-s − 6·29-s − 2·30-s − 4·31-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 − 0.485·17-s − 0.235·18-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 − 1.11·29-s − 0.365·30-s − 0.718·31-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

\(L(1)\) \(\approx\) \(1.707900010\)
\(L(\frac12)\) \(\approx\) \(1.707900010\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.42997968002786, −13.12253132319025, −12.85137877928100, −12.11589370877317, −11.53662808566706, −11.13718245173138, −10.45219182225530, −10.04103867131319, −9.635384074743526, −9.216349071534632, −8.843595620924816, −8.251027291313048, −7.581634328128506, −7.291029825718423, −6.680777810806806, −6.072670205095645, −5.724200731819376, −4.978042925119388, −4.380862487223623, −3.602735542842045, −3.100032478590056, −2.328687329916131, −2.038793951308596, −1.355891006086234, −0.4177616175489616, 0.4177616175489616, 1.355891006086234, 2.038793951308596, 2.328687329916131, 3.100032478590056, 3.602735542842045, 4.380862487223623, 4.978042925119388, 5.724200731819376, 6.072670205095645, 6.680777810806806, 7.291029825718423, 7.581634328128506, 8.251027291313048, 8.843595620924816, 9.216349071534632, 9.635384074743526, 10.04103867131319, 10.45219182225530, 11.13718245173138, 11.53662808566706, 12.11589370877317, 12.85137877928100, 13.12253132319025, 13.42997968002786

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