Properties

Label 2-236992-1.1-c1-0-40
Degree $2$
Conductor $236992$
Sign $-1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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 + 7-s + 9-s + 4·11-s − 6·13-s − 2·21-s − 5·25-s + 4·27-s − 2·29-s + 2·31-s − 8·33-s + 2·37-s + 12·39-s − 6·41-s + 4·43-s − 2·47-s + 49-s + 14·53-s − 14·59-s + 12·61-s + 63-s − 4·67-s − 2·73-s + 10·75-s + 4·77-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s − 0.436·21-s − 25-s + 0.769·27-s − 0.371·29-s + 0.359·31-s − 1.39·33-s + 0.328·37-s + 1.92·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s − 1.82·59-s + 1.53·61-s + 0.125·63-s − 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.455·77-s + 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.11800985208529, −12.27403462931609, −12.16912074002591, −11.77553911607575, −11.45959847885498, −10.88706566946013, −10.37433430260564, −9.987350798343836, −9.399107554072949, −9.118917062621225, −8.386958995544051, −7.904716026961971, −7.320898360133275, −6.908580707751320, −6.468861827093764, −5.878393735863768, −5.490106783563028, −4.961803748725557, −4.504977407101138, −4.025062789399270, −3.377992343371754, −2.578053181518633, −2.068855456724038, −1.363264240459291, −0.6716692754913710, 0, 0.6716692754913710, 1.363264240459291, 2.068855456724038, 2.578053181518633, 3.377992343371754, 4.025062789399270, 4.504977407101138, 4.961803748725557, 5.490106783563028, 5.878393735863768, 6.468861827093764, 6.908580707751320, 7.320898360133275, 7.904716026961971, 8.386958995544051, 9.118917062621225, 9.399107554072949, 9.987350798343836, 10.37433430260564, 10.88706566946013, 11.45959847885498, 11.77553911607575, 12.16912074002591, 12.27403462931609, 13.11800985208529

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