Properties

Label 2-46800-1.1-c1-0-61
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  − 5·7-s − 3·11-s + 13-s − 3·17-s + 4·19-s + 6·23-s − 9·29-s − 5·31-s + 2·37-s − 2·43-s − 9·47-s + 18·49-s + 9·53-s − 9·59-s − 61-s − 5·67-s + 14·73-s + 15·77-s + 16·79-s − 15·83-s + 6·89-s − 5·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.88·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 1.25·23-s − 1.67·29-s − 0.898·31-s + 0.328·37-s − 0.304·43-s − 1.31·47-s + 18/7·49-s + 1.23·53-s − 1.17·59-s − 0.128·61-s − 0.610·67-s + 1.63·73-s + 1.70·77-s + 1.80·79-s − 1.64·83-s + 0.635·89-s − 0.524·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + 5 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.09701103596021, −14.30676593746022, −13.50421821532044, −13.34704556954978, −12.76815843279024, −12.64125555438787, −11.69002229914577, −11.23407028985893, −10.65080784334542, −10.16475093594226, −9.560511327962515, −9.142464376936716, −8.817798709205080, −7.784054190142024, −7.443316876043353, −6.771443156900450, −6.391397984475444, −5.628068867486038, −5.275819099279802, −4.456106971418632, −3.508579602359629, −3.353857977749156, −2.622396256724056, −1.884209022916001, −0.7335600411803560, 0, 0.7335600411803560, 1.884209022916001, 2.622396256724056, 3.353857977749156, 3.508579602359629, 4.456106971418632, 5.275819099279802, 5.628068867486038, 6.391397984475444, 6.771443156900450, 7.443316876043353, 7.784054190142024, 8.817798709205080, 9.142464376936716, 9.560511327962515, 10.16475093594226, 10.65080784334542, 11.23407028985893, 11.69002229914577, 12.64125555438787, 12.76815843279024, 13.34704556954978, 13.50421821532044, 14.30676593746022, 15.09701103596021

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