Properties

Label 2-46800-1.1-c1-0-69
Degree $2$
Conductor $46800$
Sign $-1$
Analytic cond. $373.699$
Root an. cond. $19.3313$
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  − 3·7-s − 3·11-s − 13-s + 3·17-s − 4·19-s − 3·23-s − 10·29-s + 6·31-s + 5·37-s + 5·41-s + 2·43-s + 2·47-s + 2·49-s + 11·53-s − 4·59-s + 61-s − 4·67-s − 3·71-s − 6·73-s + 9·77-s − 3·79-s + 16·83-s + 7·89-s + 3·91-s + 19·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.13·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.625·23-s − 1.85·29-s + 1.07·31-s + 0.821·37-s + 0.780·41-s + 0.304·43-s + 0.291·47-s + 2/7·49-s + 1.51·53-s − 0.520·59-s + 0.128·61-s − 0.488·67-s − 0.356·71-s − 0.702·73-s + 1.02·77-s − 0.337·79-s + 1.75·83-s + 0.741·89-s + 0.314·91-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(373.699\)
Root analytic conductor: \(19.3313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{46800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−14.80933389582634, −14.52635306610262, −13.58342273566513, −13.29662191300666, −12.90281712510746, −12.30020155138515, −11.91898420595482, −11.19847172457691, −10.49018039521694, −10.30983991707860, −9.567454513233203, −9.278136488414308, −8.552791800126393, −7.818440116485434, −7.571323686238594, −6.835014518629006, −6.164758127465801, −5.807202235273064, −5.194033494798452, −4.354398896778382, −3.862986635212327, −3.120773720167048, −2.557483812033145, −1.931763830659196, −0.7691244434083201, 0, 0.7691244434083201, 1.931763830659196, 2.557483812033145, 3.120773720167048, 3.862986635212327, 4.354398896778382, 5.194033494798452, 5.807202235273064, 6.164758127465801, 6.835014518629006, 7.571323686238594, 7.818440116485434, 8.552791800126393, 9.278136488414308, 9.567454513233203, 10.30983991707860, 10.49018039521694, 11.19847172457691, 11.91898420595482, 12.30020155138515, 12.90281712510746, 13.29662191300666, 13.58342273566513, 14.52635306610262, 14.80933389582634

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