Properties

Label 2-1200-15.14-c2-0-5
Degree $2$
Conductor $1200$
Sign $-0.447 - 0.894i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 13i·7-s − 9·9-s − 23i·13-s + 11·19-s + 39·21-s + 27i·27-s − 59·31-s + 26i·37-s − 69·39-s + 83i·43-s − 120·49-s − 33i·57-s − 121·61-s − 117i·63-s + ⋯
L(s)  = 1  i·3-s + 1.85i·7-s − 9-s − 1.76i·13-s + 0.578·19-s + 1.85·21-s + i·27-s − 1.90·31-s + 0.702i·37-s − 1.76·39-s + 1.93i·43-s − 2.44·49-s − 0.578i·57-s − 1.98·61-s − 1.85i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5414333018\)
\(L(\frac12)\) \(\approx\) \(0.5414333018\)
\(L(2)\) 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 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 - 13iT - 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 23iT - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 - 11T + 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 59T + 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 83iT - 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + 121T + 3.72e3T^{2} \)
67 \( 1 - 13iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 46iT - 5.32e3T^{2} \)
79 \( 1 + 142T + 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 - 167iT - 9.40e3T^{2} \)
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

−9.596125079566744108661686525046, −8.881024066759561667128452120861, −8.114061847201436409623374491730, −7.54488443915516510283392574198, −6.32804281240571736527096397442, −5.66828971538788621540513019141, −5.13650467608733987479659882388, −3.19981813948027966126231581276, −2.62016294013058508604226690154, −1.42332371151406899348358766287, 0.15677652851306412910321358092, 1.74377695506693417430781020040, 3.41512080014130764067236388147, 4.07092034965117253693955869920, 4.69938630315114602413932672281, 5.83606009752520577561577590139, 7.00816169936187185627896678697, 7.44603916369347235577420959500, 8.728600786314345023625280564125, 9.389777395671716201513469034411

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