Properties

Label 2-116886-1.1-c1-0-45
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 $2$

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 − 2·13-s + 14-s + 16-s − 6·17-s − 18-s − 2·19-s − 21-s + 23-s − 24-s − 5·25-s + 2·26-s + 27-s − 28-s − 31-s − 32-s + 6·34-s + 36-s − 4·37-s + 2·38-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 − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.179·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.657·37-s + 0.324·38-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: \(2\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.99969740760010, −13.56608220043192, −13.11117429053334, −12.73323052416592, −12.04096235416897, −11.59832491189644, −11.16498842580830, −10.49106957671372, −10.12759511933988, −9.575495348186614, −9.264181429663331, −8.603220225397512, −8.277363191314317, −7.811932683261328, −7.072915237766622, −6.683470979391236, −6.390445844098919, −5.536009891458023, −4.860289912777990, −4.434184495654539, −3.495404359946009, −3.292751321467969, −2.356010590358272, −1.987839561141296, −1.378983421847602, 0, 0, 1.378983421847602, 1.987839561141296, 2.356010590358272, 3.292751321467969, 3.495404359946009, 4.434184495654539, 4.860289912777990, 5.536009891458023, 6.390445844098919, 6.683470979391236, 7.072915237766622, 7.811932683261328, 8.277363191314317, 8.603220225397512, 9.264181429663331, 9.575495348186614, 10.12759511933988, 10.49106957671372, 11.16498842580830, 11.59832491189644, 12.04096235416897, 12.73323052416592, 13.11117429053334, 13.56608220043192, 13.99969740760010

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