Properties

Label 2-46800-1.1-c1-0-70
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 4·11-s + 13-s + 8·17-s + 6·19-s − 6·23-s + 4·29-s + 2·37-s + 2·41-s − 4·43-s − 3·49-s − 10·53-s + 4·59-s − 10·61-s + 12·67-s − 8·71-s + 8·73-s + 8·77-s − 8·79-s − 12·83-s + 14·89-s + 2·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.20·11-s + 0.277·13-s + 1.94·17-s + 1.37·19-s − 1.25·23-s + 0.742·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 3/7·49-s − 1.37·53-s + 0.520·59-s − 1.28·61-s + 1.46·67-s − 0.949·71-s + 0.936·73-s + 0.911·77-s − 0.900·79-s − 1.31·83-s + 1.48·89-s + 0.209·91-s + 1.62·97-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.814397846\)
\(L(\frac12)\) \(\approx\) \(3.814397846\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.44410015859292, −14.10382747394510, −13.91095028434095, −13.02819656188080, −12.36434839820494, −11.96046643923441, −11.60862413080823, −11.13830300197299, −10.34874278291171, −9.765876624046945, −9.597674716993578, −8.732574674214248, −8.239453528792638, −7.664654454646305, −7.360959053779086, −6.410232911643341, −6.043697964969649, −5.373250677828592, −4.818735079381951, −4.147286172182267, −3.438391635772257, −3.075684303389835, −1.923198472222481, −1.357694032103301, −0.7747762876970657, 0.7747762876970657, 1.357694032103301, 1.923198472222481, 3.075684303389835, 3.438391635772257, 4.147286172182267, 4.818735079381951, 5.373250677828592, 6.043697964969649, 6.410232911643341, 7.360959053779086, 7.664654454646305, 8.239453528792638, 8.732574674214248, 9.597674716993578, 9.765876624046945, 10.34874278291171, 11.13830300197299, 11.60862413080823, 11.96046643923441, 12.36434839820494, 13.02819656188080, 13.91095028434095, 14.10382747394510, 14.44410015859292

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