↑

Properties

Label 2-1200-5.2-c2-0-13
Degree $2$
Conductor $1200$
Sign $-0.229 - 0.973i$
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  + (−1.22 + 1.22i)3-s + (7.44 + 7.44i)7-s − 2.99i·9-s + 16.2·11-s + (−12.2 + 12.2i)13-s + (7.55 + 7.55i)17-s + 14.4i·19-s − 18.2·21-s + (−2.65 + 2.65i)23-s + (3.67 + 3.67i)27-s − 34.2i·29-s + 20.4·31-s + (−19.8 + 19.8i)33-s + (−7.34 − 7.34i)37-s − 29.9i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (1.06 + 1.06i)7-s − 0.333i·9-s + 1.47·11-s + (−0.942 + 0.942i)13-s + (0.444 + 0.444i)17-s + 0.762i·19-s − 0.868·21-s + (−0.115 + 0.115i)23-s + (0.136 + 0.136i)27-s − 1.18i·29-s + 0.661·31-s + (−0.602 + 0.602i)33-s + (−0.198 − 0.198i)37-s − 0.769i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.903278139\)
\(L(\frac12)\) \(\approx\) \(1.903278139\)
\(L(2)\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-7.44 - 7.44i)T + 49iT^{2} \)
11 \( 1 - 16.2T + 121T^{2} \)
13 \( 1 + (12.2 - 12.2i)T - 169iT^{2} \)
17 \( 1 + (-7.55 - 7.55i)T + 289iT^{2} \)
19 \( 1 - 14.4iT - 361T^{2} \)
23 \( 1 + (2.65 - 2.65i)T - 529iT^{2} \)
29 \( 1 + 34.2iT - 841T^{2} \)
31 \( 1 - 20.4T + 961T^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 - 25.5T + 1.68e3T^{2} \)
43 \( 1 + (-25.1 + 25.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-22.0 - 22.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-35.3 + 35.3i)T - 2.80e3iT^{2} \)
59 \( 1 - 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (24.6 + 24.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 77.9T + 5.04e3T^{2} \)
73 \( 1 + (-44.1 + 44.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 48.4iT - 6.24e3T^{2} \)
83 \( 1 + (101. - 101. i)T - 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (55.4 + 55.4i)T + 9.40e3iT^{2} \)
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

−9.632859909162151747613119616598, −9.104732284596143137377796379596, −8.277698791076276690468652600707, −7.33384624441647957060633683590, −6.24183557027119255213741613547, −5.62876967100398887823409960853, −4.57066846018239856135781362346, −3.95127075558606183119643194232, −2.38784960880184873483644749360, −1.37034183435050112049221645310, 0.66238431173483204100224253549, 1.53370119436264792630276150082, 3.01420071208612756990260548904, 4.31652864056543687701961546236, 4.92894297191316099984527433320, 5.98984847145059838005657110661, 7.10578896838028352919498473974, 7.41950978254301261027165202197, 8.392664110340091837186598221485, 9.342543125851746256120295937140

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