Properties

Label 2-209e2-1.1-c1-0-11
Degree $2$
Conductor $43681$
Sign $-1$
Analytic cond. $348.794$
Root an. cond. $18.6760$
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  + 2·3-s − 2·4-s + 3·5-s + 7-s + 9-s − 4·12-s − 4·13-s + 6·15-s + 4·16-s + 3·17-s − 6·20-s + 2·21-s + 4·25-s − 4·27-s − 2·28-s + 6·29-s + 4·31-s + 3·35-s − 2·36-s − 2·37-s − 8·39-s − 6·41-s + 43-s + 3·45-s − 3·47-s + 8·48-s − 6·49-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 1.15·12-s − 1.10·13-s + 1.54·15-s + 16-s + 0.727·17-s − 1.34·20-s + 0.436·21-s + 4/5·25-s − 0.769·27-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.507·35-s − 1/3·36-s − 0.328·37-s − 1.28·39-s − 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.437·47-s + 1.15·48-s − 6/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(43681\)    =    \(11^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(348.794\)
Root analytic conductor: \(18.6760\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 43681,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.66316846801476, −14.33856682119376, −13.96109575948699, −13.54312919204235, −13.16507451758571, −12.39410251777930, −12.15802499208649, −11.24913908111154, −10.46818811105109, −9.922012831940696, −9.599236391319428, −9.367802799266532, −8.448858453902896, −8.285458397428332, −7.751643530904382, −6.912407972246272, −6.322973896061977, −5.453096261840122, −5.203115410959213, −4.541959481460336, −3.839053925215717, −2.972290989907395, −2.692989716240768, −1.791970373083437, −1.246228753992631, 0, 1.246228753992631, 1.791970373083437, 2.692989716240768, 2.972290989907395, 3.839053925215717, 4.541959481460336, 5.203115410959213, 5.453096261840122, 6.322973896061977, 6.912407972246272, 7.751643530904382, 8.285458397428332, 8.448858453902896, 9.367802799266532, 9.599236391319428, 9.922012831940696, 10.46818811105109, 11.24913908111154, 12.15802499208649, 12.39410251777930, 13.16507451758571, 13.54312919204235, 13.96109575948699, 14.33856682119376, 14.66316846801476

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