↑

Properties

Label 2-1200-15.2-c1-0-24
Degree $2$
Conductor $1200$
Sign $0.845 + 0.533i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.912 + 1.47i)3-s + (−1.78 − 1.78i)7-s + (−1.33 + 2.68i)9-s − 4.25i·11-s + (2.90 − 2.90i)13-s + (−0.443 + 0.443i)17-s − 7.74i·19-s + (1 − 4.25i)21-s + (1.94 + 1.94i)23-s + (−5.17 + 0.483i)27-s + 10.0·29-s − 0.372·31-s + (6.26 − 3.88i)33-s + (1.12 + 1.12i)37-s + (6.92 + 1.62i)39-s + ⋯
L(s)  = 1  + (0.526 + 0.850i)3-s + (−0.674 − 0.674i)7-s + (−0.445 + 0.895i)9-s − 1.28i·11-s + (0.805 − 0.805i)13-s + (−0.107 + 0.107i)17-s − 1.77i·19-s + (0.218 − 0.928i)21-s + (0.404 + 0.404i)23-s + (−0.995 + 0.0929i)27-s + 1.87·29-s − 0.0668·31-s + (1.09 − 0.675i)33-s + (0.184 + 0.184i)37-s + (1.10 + 0.260i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.845 + 0.533i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.691369853\)
\(L(\frac12)\) \(\approx\) \(1.691369853\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.912 - 1.47i)T \)
5 \( 1 \)
good7 \( 1 + (1.78 + 1.78i)T + 7iT^{2} \)
11 \( 1 + 4.25iT - 11T^{2} \)
13 \( 1 + (-2.90 + 2.90i)T - 13iT^{2} \)
17 \( 1 + (0.443 - 0.443i)T - 17iT^{2} \)
19 \( 1 + 7.74iT - 19T^{2} \)
23 \( 1 + (-1.94 - 1.94i)T + 23iT^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + 0.372T + 31T^{2} \)
37 \( 1 + (-1.12 - 1.12i)T + 37iT^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + (6.68 - 6.68i)T - 43iT^{2} \)
47 \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \)
53 \( 1 + (-7.59 - 7.59i)T + 53iT^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 4.37T + 61T^{2} \)
67 \( 1 + (1.22 + 1.22i)T + 67iT^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + (7.90 - 7.90i)T - 73iT^{2} \)
79 \( 1 - 2.74iT - 79T^{2} \)
83 \( 1 + (-8.04 - 8.04i)T + 83iT^{2} \)
89 \( 1 + 0.497T + 89T^{2} \)
97 \( 1 + (6.47 + 6.47i)T + 97iT^{2} \)
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

−9.659092391690235026487415961988, −8.786394249873002779917748412909, −8.365866655433518386517419102816, −7.24104989845530997254165817846, −6.29568004185642777219062279387, −5.35053312247263364277342931225, −4.35562993882839235088833731237, −3.35086797506525573984825573372, −2.83046980340869994701147314280, −0.72275844543377487449831276911, 1.44787687193921142344397536752, 2.43590775972196229579703053410, 3.48581794239602393495843517635, 4.56866423069954201981510745548, 5.96221807611289813065089513897, 6.50436765683536628107514861972, 7.30963822298788881001538570488, 8.262623754487133212311810207057, 8.900281946953818300495490118752, 9.693646297974380694728642734553

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