Properties

Label 2-116886-1.1-c1-0-39
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 + 3·5-s − 6-s + 7-s − 8-s + 9-s − 3·10-s + 12-s + 3·13-s − 14-s + 3·15-s + 16-s + 4·17-s − 18-s + 3·20-s + 21-s + 23-s − 24-s + 4·25-s − 3·26-s + 27-s + 28-s − 3·29-s − 3·30-s − 6·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.670·20-s + 0.218·21-s + 0.208·23-s − 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.547·30-s − 1.07·31-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.82465487579594, −13.52575431764444, −12.76796393019971, −12.61433558366622, −11.79154434976217, −11.27636476848274, −10.81592877112736, −10.10233141353036, −10.06668367850559, −9.409496402524015, −8.785327011211012, −8.733130341348566, −7.914716429245145, −7.557536164736276, −6.820034323475776, −6.492431191709498, −5.673895734468614, −5.477591071308801, −4.821674108294610, −3.875989311111214, −3.369991583952398, −2.821417625984359, −1.973069133426014, −1.634868308058267, −1.180364558894246, 0, 1.180364558894246, 1.634868308058267, 1.973069133426014, 2.821417625984359, 3.369991583952398, 3.875989311111214, 4.821674108294610, 5.477591071308801, 5.673895734468614, 6.492431191709498, 6.820034323475776, 7.557536164736276, 7.914716429245145, 8.733130341348566, 8.785327011211012, 9.409496402524015, 10.06668367850559, 10.10233141353036, 10.81592877112736, 11.27636476848274, 11.79154434976217, 12.61433558366622, 12.76796393019971, 13.52575431764444, 13.82465487579594

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