Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 9-s − 4·11-s − 6·17-s + 6·19-s + 2·21-s − 5·25-s − 4·27-s − 10·29-s − 4·31-s − 8·33-s − 2·37-s − 10·41-s + 4·43-s − 12·47-s + 49-s − 12·51-s − 6·53-s + 12·57-s − 2·59-s + 63-s + 8·71-s − 6·73-s − 10·75-s − 4·77-s − 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 25-s − 0.769·27-s − 1.85·29-s − 0.718·31-s − 1.39·33-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s − 0.824·53-s + 1.58·57-s − 0.260·59-s + 0.125·63-s + 0.949·71-s − 0.702·73-s − 1.15·75-s − 0.455·77-s − 0.900·79-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: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4305957804\)
\(L(\frac12)\) \(\approx\) \(0.4305957804\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.18731563654872, −12.61546952543790, −11.84713721143045, −11.53863323792777, −10.97701106183263, −10.71335471855975, −9.825459233530456, −9.694091325732777, −9.023764299519528, −8.804748654394443, −8.061686106107468, −7.787208243405352, −7.501068533565237, −6.850925816058896, −6.260614160857431, −5.476242895021765, −5.266595721670389, −4.674153763516817, −3.929261681787807, −3.480126922031294, −3.050958641884696, −2.330260066376909, −1.966103143986144, −1.455518542436019, −0.1495640277841021, 0.1495640277841021, 1.455518542436019, 1.966103143986144, 2.330260066376909, 3.050958641884696, 3.480126922031294, 3.929261681787807, 4.674153763516817, 5.266595721670389, 5.476242895021765, 6.260614160857431, 6.850925816058896, 7.501068533565237, 7.787208243405352, 8.061686106107468, 8.804748654394443, 9.023764299519528, 9.694091325732777, 9.825459233530456, 10.71335471855975, 10.97701106183263, 11.53863323792777, 11.84713721143045, 12.61546952543790, 13.18731563654872

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