Properties

Label 2-236992-1.1-c1-0-21
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-s − 4·5-s + 7-s − 2·9-s − 2·13-s − 4·15-s − 5·17-s − 7·19-s + 21-s + 11·25-s − 5·27-s − 8·29-s − 2·31-s − 4·35-s + 8·37-s − 2·39-s − 2·41-s − 3·43-s + 8·45-s + 49-s − 5·51-s − 6·53-s − 7·57-s + 9·59-s + 6·61-s − 2·63-s + 8·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 0.377·7-s − 2/3·9-s − 0.554·13-s − 1.03·15-s − 1.21·17-s − 1.60·19-s + 0.218·21-s + 11/5·25-s − 0.962·27-s − 1.48·29-s − 0.359·31-s − 0.676·35-s + 1.31·37-s − 0.320·39-s − 0.312·41-s − 0.457·43-s + 1.19·45-s + 1/7·49-s − 0.700·51-s − 0.824·53-s − 0.927·57-s + 1.17·59-s + 0.768·61-s − 0.251·63-s + 0.992·65-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 \( 1 - T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.99899407844726, −12.78589406059965, −12.15372631696678, −11.59903152736709, −11.30087678238648, −11.00917933414309, −10.57874558782377, −9.759861417362779, −9.232898274017929, −8.799172764433933, −8.307510551312660, −8.088469301572770, −7.621931769002018, −7.046441578623236, −6.651650607424321, −6.041836140314664, −5.242893995510662, −4.852554569591972, −4.109211769489709, −3.971173245758483, −3.441647568868886, −2.547199582675381, −2.394208910478221, −1.571222723305494, −0.4738242234869243, 0, 0.4738242234869243, 1.571222723305494, 2.394208910478221, 2.547199582675381, 3.441647568868886, 3.971173245758483, 4.109211769489709, 4.852554569591972, 5.242893995510662, 6.041836140314664, 6.651650607424321, 7.046441578623236, 7.621931769002018, 8.088469301572770, 8.307510551312660, 8.799172764433933, 9.232898274017929, 9.759861417362779, 10.57874558782377, 11.00917933414309, 11.30087678238648, 11.59903152736709, 12.15372631696678, 12.78589406059965, 12.99899407844726

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