Properties

Label 2-46800-1.1-c1-0-92
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 + 4·11-s − 13-s + 2·17-s + 10·29-s − 4·31-s + 2·37-s − 6·41-s − 12·43-s + 9·49-s + 6·53-s + 12·59-s − 2·61-s − 8·67-s − 2·73-s − 16·77-s − 8·79-s − 4·83-s + 2·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.20·11-s − 0.277·13-s + 0.485·17-s + 1.85·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.82·43-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s − 0.977·67-s − 0.234·73-s − 1.82·77-s − 0.900·79-s − 0.439·83-s + 0.211·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.75859499202109, −14.42411154772300, −13.75899233536595, −13.29697691936743, −12.84075962532412, −12.21385809623017, −11.84390293864844, −11.46124814464235, −10.41031453840180, −10.18756428404471, −9.694218947993799, −9.128163101218480, −8.651968496854667, −8.078281701434993, −7.198110086251223, −6.745188623544328, −6.462500634744769, −5.774193023517916, −5.145902072596059, −4.354564554568270, −3.762592238951736, −3.202530659678546, −2.691456692193998, −1.703542873373594, −0.9195064949044537, 0, 0.9195064949044537, 1.703542873373594, 2.691456692193998, 3.202530659678546, 3.762592238951736, 4.354564554568270, 5.145902072596059, 5.774193023517916, 6.462500634744769, 6.745188623544328, 7.198110086251223, 8.078281701434993, 8.651968496854667, 9.128163101218480, 9.694218947993799, 10.18756428404471, 10.41031453840180, 11.46124814464235, 11.84390293864844, 12.21385809623017, 12.84075962532412, 13.29697691936743, 13.75899233536595, 14.42411154772300, 14.75859499202109

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