Properties

Label 2-236992-1.1-c1-0-17
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 + 4·13-s − 6·17-s + 2·19-s − 2·21-s − 5·25-s − 4·27-s + 6·29-s − 4·31-s + 2·37-s + 8·39-s + 6·41-s + 8·43-s − 12·47-s + 49-s − 12·51-s + 6·53-s + 4·57-s + 6·59-s + 8·61-s − 63-s − 4·67-s + 2·73-s − 10·75-s − 8·79-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 0.328·37-s + 1.28·39-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 1.68·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s + 1.02·61-s − 0.125·63-s − 0.488·67-s + 0.234·73-s − 1.15·75-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

\(L(1)\) \(\approx\) \(3.126928339\)
\(L(\frac12)\) \(\approx\) \(3.126928339\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.12917195101859, −12.73659383495597, −11.87104041272984, −11.35204021519883, −11.28480321790200, −10.43117831481402, −10.09697083181932, −9.493692497204458, −9.004114579819869, −8.791989974446936, −8.317123806205391, −7.718176324562899, −7.432752371940186, −6.627372272243193, −6.345287771861489, −5.773697051546522, −5.200276543673204, −4.455770860890105, −3.868509578320793, −3.704799121705127, −2.877484887540865, −2.531090506649586, −1.951268525630507, −1.268660890805920, −0.4442676835914295, 0.4442676835914295, 1.268660890805920, 1.951268525630507, 2.531090506649586, 2.877484887540865, 3.704799121705127, 3.868509578320793, 4.455770860890105, 5.200276543673204, 5.773697051546522, 6.345287771861489, 6.627372272243193, 7.432752371940186, 7.718176324562899, 8.317123806205391, 8.791989974446936, 9.004114579819869, 9.493692497204458, 10.09697083181932, 10.43117831481402, 11.28480321790200, 11.35204021519883, 11.87104041272984, 12.73659383495597, 13.12917195101859

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