Properties

Label 2-46800-1.1-c1-0-64
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  − 4·7-s − 2·11-s + 13-s − 6·17-s + 4·19-s − 4·23-s − 10·29-s + 8·31-s + 2·37-s − 4·43-s − 2·47-s + 9·49-s − 2·53-s + 10·59-s + 10·61-s + 8·67-s + 2·71-s + 10·73-s + 8·77-s − 8·79-s − 6·83-s + 12·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.603·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s − 1.85·29-s + 1.43·31-s + 0.328·37-s − 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.274·53-s + 1.30·59-s + 1.28·61-s + 0.977·67-s + 0.237·71-s + 1.17·73-s + 0.911·77-s − 0.900·79-s − 0.658·83-s + 1.27·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.92510725225134, −14.31764830285277, −13.57956290813401, −13.23062030509142, −13.05957606459966, −12.34039964851241, −11.75394337276787, −11.24585113831891, −10.72415428258314, −9.978350152987954, −9.704686140427164, −9.263835737661182, −8.501257767453184, −8.075167726604508, −7.321011956423130, −6.743992448766432, −6.407989106078661, −5.687718429883635, −5.235951665020738, −4.345552561455069, −3.786727840035262, −3.215887440863487, −2.521650309440568, −1.945776512239090, −0.7458440316785662, 0, 0.7458440316785662, 1.945776512239090, 2.521650309440568, 3.215887440863487, 3.786727840035262, 4.345552561455069, 5.235951665020738, 5.687718429883635, 6.407989106078661, 6.743992448766432, 7.321011956423130, 8.075167726604508, 8.501257767453184, 9.263835737661182, 9.704686140427164, 9.978350152987954, 10.72415428258314, 11.24585113831891, 11.75394337276787, 12.34039964851241, 13.05957606459966, 13.23062030509142, 13.57956290813401, 14.31764830285277, 14.92510725225134

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