Properties

Label 2-116886-1.1-c1-0-25
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 + 13-s + 14-s + 16-s + 3·17-s − 18-s − 5·19-s − 21-s − 23-s − 24-s − 5·25-s − 26-s + 27-s − 28-s − 9·29-s + 5·31-s − 32-s − 3·34-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 + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.208·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-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 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.79874955284284, −13.35583857745434, −12.89688665656316, −12.29455969522033, −11.99586598648521, −11.29320475855965, −10.74766932978984, −10.44711746826096, −9.739993312121955, −9.493003205102358, −8.897899637433212, −8.534517643262872, −7.825285492037044, −7.581369130525841, −7.062895556517878, −6.262361333294018, −5.983261079515614, −5.415227752242244, −4.504543356262086, −3.872609291692726, −3.606774702499376, −2.628142800989755, −2.342152901747529, −1.586763476538596, −0.8538505861213762, 0, 0.8538505861213762, 1.586763476538596, 2.342152901747529, 2.628142800989755, 3.606774702499376, 3.872609291692726, 4.504543356262086, 5.415227752242244, 5.983261079515614, 6.262361333294018, 7.062895556517878, 7.581369130525841, 7.825285492037044, 8.534517643262872, 8.897899637433212, 9.493003205102358, 9.739993312121955, 10.44711746826096, 10.74766932978984, 11.29320475855965, 11.99586598648521, 12.29455969522033, 12.89688665656316, 13.35583857745434, 13.79874955284284

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