Properties

Label 2-46800-1.1-c1-0-28
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s + 4·11-s − 13-s − 6·17-s + 8·23-s − 6·29-s + 4·31-s + 2·37-s + 10·41-s − 4·43-s + 8·47-s + 9·49-s − 2·53-s + 12·59-s − 2·61-s − 16·67-s − 8·71-s + 6·73-s − 16·77-s + 16·79-s − 4·83-s + 2·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s − 0.277·13-s − 1.45·17-s + 1.66·23-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.274·53-s + 1.56·59-s − 0.256·61-s − 1.95·67-s − 0.949·71-s + 0.702·73-s − 1.82·77-s + 1.80·79-s − 0.439·83-s + 0.211·89-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.546219085\)
\(L(\frac12)\) \(\approx\) \(1.546219085\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.83921643834328, −13.98920747277382, −13.42012097660552, −13.10643406873107, −12.65906351931923, −12.02843938186356, −11.56163549859028, −10.88834875039000, −10.56606852463115, −9.682831341779686, −9.267842163194186, −9.110365951198390, −8.439330436171774, −7.516534891989987, −6.972450672198610, −6.643919891080329, −6.114202388363517, −5.518920831590224, −4.631720087448510, −4.126343546598230, −3.538372753467111, −2.835016226323715, −2.312234398821373, −1.273140152225592, −0.4675342999923367, 0.4675342999923367, 1.273140152225592, 2.312234398821373, 2.835016226323715, 3.538372753467111, 4.126343546598230, 4.631720087448510, 5.518920831590224, 6.114202388363517, 6.643919891080329, 6.972450672198610, 7.516534891989987, 8.439330436171774, 9.110365951198390, 9.267842163194186, 9.682831341779686, 10.56606852463115, 10.88834875039000, 11.56163549859028, 12.02843938186356, 12.65906351931923, 13.10643406873107, 13.42012097660552, 13.98920747277382, 14.83921643834328

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