Properties

Label 2-236992-1.1-c1-0-51
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 7-s + 6·9-s − 6·13-s − 7·17-s − 5·19-s + 3·21-s − 5·25-s − 9·27-s − 8·29-s + 2·31-s + 4·37-s + 18·39-s + 6·41-s + 3·43-s − 8·47-s + 49-s + 21·51-s + 6·53-s + 15·57-s + 5·59-s + 10·61-s − 6·63-s − 13·67-s + 6·71-s − 11·73-s + 15·75-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s − 1.66·13-s − 1.69·17-s − 1.14·19-s + 0.654·21-s − 25-s − 1.73·27-s − 1.48·29-s + 0.359·31-s + 0.657·37-s + 2.88·39-s + 0.937·41-s + 0.457·43-s − 1.16·47-s + 1/7·49-s + 2.94·51-s + 0.824·53-s + 1.98·57-s + 0.650·59-s + 1.28·61-s − 0.755·63-s − 1.58·67-s + 0.712·71-s − 1.28·73-s + 1.73·75-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: \(2\)
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 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.04551821777950, −12.89034013093946, −12.50154656301288, −11.68067044913385, −11.60528115309008, −11.25447127322265, −10.50089569134192, −10.29983752623886, −9.817948545745276, −9.232852892579119, −8.891988193786219, −8.065039102302018, −7.402809728348490, −7.203407191098024, −6.564804596920823, −6.206473176148721, −5.777245946399980, −5.205567683446030, −4.718514773027870, −4.219456484490143, −3.971153116927906, −2.888781489197433, −2.161313927783363, −1.928843974891282, −0.8330212057667435, 0, 0, 0.8330212057667435, 1.928843974891282, 2.161313927783363, 2.888781489197433, 3.971153116927906, 4.219456484490143, 4.718514773027870, 5.205567683446030, 5.777245946399980, 6.206473176148721, 6.564804596920823, 7.203407191098024, 7.402809728348490, 8.065039102302018, 8.891988193786219, 9.232852892579119, 9.817948545745276, 10.29983752623886, 10.50089569134192, 11.25447127322265, 11.60528115309008, 11.68067044913385, 12.50154656301288, 12.89034013093946, 13.04551821777950

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