↑

Properties

Label 2-2400-600.509-c0-0-0
Degree $2$
Conductor $2400$
Sign $0.187 + 0.982i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)3-s + (0.587 − 0.809i)5-s − 1.90i·7-s + (0.809 − 0.587i)9-s + (0.951 + 0.690i)11-s + (−0.309 + 0.951i)15-s + (0.587 + 1.80i)21-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (0.190 − 0.587i)31-s + (−1.11 − 0.363i)33-s + (−1.53 − 1.11i)35-s − 0.999i·45-s − 2.61·49-s + (−1.53 + 0.5i)53-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s + (0.587 − 0.809i)5-s − 1.90i·7-s + (0.809 − 0.587i)9-s + (0.951 + 0.690i)11-s + (−0.309 + 0.951i)15-s + (0.587 + 1.80i)21-s + (−0.309 − 0.951i)25-s + (−0.587 + 0.809i)27-s + (0.190 − 0.587i)31-s + (−1.11 − 0.363i)33-s + (−1.53 − 1.11i)35-s − 0.999i·45-s − 2.61·49-s + (−1.53 + 0.5i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9959245315\)
\(L(\frac12)\) \(\approx\) \(0.9959245315\)
\(L(1)\) 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 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
good7 \( 1 + 1.90iT - T^{2} \)
11 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.80 + 0.587i)T + (0.809 - 0.587i)T^{2} \)
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

−9.323585860522308747777126667861, −8.145162447865660695502569608065, −7.22762039199995340578980592815, −6.65966277005694332814700263432, −5.89628183986371464617539175157, −4.80128533014847464032958892155, −4.37349597785914493795188421582, −3.63364707751390948912127311164, −1.69521048608441754662699096668, −0.826995455620520213626244731468, 1.58194238199861338136661776316, 2.49056516009425492262741995253, 3.48087763550110765220193362377, 4.89542967713848007246460245014, 5.61051294231334126005489771807, 6.24681734452838046943762379747, 6.60203453980879342351895294684, 7.70014646931068056856934425457, 8.696471273857746835868428188614, 9.288967744747587389590789023038

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