Properties

Label 2-236992-1.1-c1-0-43
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  − 3·3-s − 7-s + 6·9-s + 6·11-s − 13-s + 3·21-s − 5·25-s − 9·27-s + 3·29-s + 3·31-s − 18·33-s − 8·37-s + 3·39-s + 9·41-s − 4·43-s − 13·47-s + 49-s + 4·53-s + 4·59-s + 2·61-s − 6·63-s + 4·67-s + 5·71-s + 3·73-s + 15·75-s − 6·77-s + 12·79-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 0.654·21-s − 25-s − 1.73·27-s + 0.557·29-s + 0.538·31-s − 3.13·33-s − 1.31·37-s + 0.480·39-s + 1.40·41-s − 0.609·43-s − 1.89·47-s + 1/7·49-s + 0.549·53-s + 0.520·59-s + 0.256·61-s − 0.755·63-s + 0.488·67-s + 0.593·71-s + 0.351·73-s + 1.73·75-s − 0.683·77-s + 1.35·79-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 \( 1 + p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.92912426423835, −12.52442259653859, −12.00940603669233, −11.80721383629319, −11.37870511661205, −10.99288643468298, −10.34556362699779, −9.933874737401427, −9.596271657787141, −9.092719230841963, −8.440598288304115, −7.898864532934102, −7.127679317193222, −6.815622902031324, −6.415843920009288, −6.080163123320778, −5.481565664965109, −5.004389592573105, −4.487855722383146, −3.885186770835277, −3.614623687261611, −2.673218102650663, −1.823668496685330, −1.315920440558862, −0.7100256461647106, 0, 0.7100256461647106, 1.315920440558862, 1.823668496685330, 2.673218102650663, 3.614623687261611, 3.885186770835277, 4.487855722383146, 5.004389592573105, 5.481565664965109, 6.080163123320778, 6.415843920009288, 6.815622902031324, 7.127679317193222, 7.898864532934102, 8.440598288304115, 9.092719230841963, 9.596271657787141, 9.933874737401427, 10.34556362699779, 10.99288643468298, 11.37870511661205, 11.80721383629319, 12.00940603669233, 12.52442259653859, 12.92912426423835

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