↑

Properties

Label 2-600-5.3-c2-0-8
Degree $2$
Conductor $600$
Sign $0.608 - 0.793i$
Analytic cond. $16.3488$
Root an. cond. $4.04336$
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 + (0.775 − 0.775i)7-s + 2.99i·9-s + 2.89·11-s + (5.87 + 5.87i)13-s + (4.44 − 4.44i)17-s − 0.101i·19-s + 1.89·21-s + (25.3 + 25.3i)23-s + (−3.67 + 3.67i)27-s + 32.2i·29-s − 3.69·31-s + (3.55 + 3.55i)33-s + (42.6 − 42.6i)37-s + 14.3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.110 − 0.110i)7-s + 0.333i·9-s + 0.263·11-s + (0.452 + 0.452i)13-s + (0.261 − 0.261i)17-s − 0.00531i·19-s + 0.0904·21-s + (1.10 + 1.10i)23-s + (−0.136 + 0.136i)27-s + 1.11i·29-s − 0.119·31-s + (0.107 + 0.107i)33-s + (1.15 − 1.15i)37-s + 0.369i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(16.3488\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1),\ 0.608 - 0.793i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.153900252\)
\(L(\frac12)\) \(\approx\) \(2.153900252\)
\(L(2)\) 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 \( 1 + (-0.775 + 0.775i)T - 49iT^{2} \)
11 \( 1 - 2.89T + 121T^{2} \)
13 \( 1 + (-5.87 - 5.87i)T + 169iT^{2} \)
17 \( 1 + (-4.44 + 4.44i)T - 289iT^{2} \)
19 \( 1 + 0.101iT - 361T^{2} \)
23 \( 1 + (-25.3 - 25.3i)T + 529iT^{2} \)
29 \( 1 - 32.2iT - 841T^{2} \)
31 \( 1 + 3.69T + 961T^{2} \)
37 \( 1 + (-42.6 + 42.6i)T - 1.36e3iT^{2} \)
41 \( 1 + 12.8T + 1.68e3T^{2} \)
43 \( 1 + (-49.2 - 49.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-2.85 + 2.85i)T - 2.20e3iT^{2} \)
53 \( 1 + (13.1 + 13.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 76.3iT - 3.48e3T^{2} \)
61 \( 1 + 103.T + 3.72e3T^{2} \)
67 \( 1 + (-47.6 + 47.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 29.7T + 5.04e3T^{2} \)
73 \( 1 + (-3.50 - 3.50i)T + 5.32e3iT^{2} \)
79 \( 1 + 87.7iT - 6.24e3T^{2} \)
83 \( 1 + (-81.7 - 81.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 96.5iT - 7.92e3T^{2} \)
97 \( 1 + (-54.2 + 54.2i)T - 9.40e3iT^{2} \)
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

−10.72859302123912347981627690782, −9.386271485676022805314637455659, −9.158920161841534376789161208601, −7.934405188335807505961091961328, −7.15608564776996722058021381486, −5.99707032142572137861563293168, −4.92316549027702990729327825434, −3.90695098922698027938242351174, −2.87581195288427901548100638631, −1.33936140956052270359159840803, 0.892343155790627957989094404399, 2.35242428200595951349936936704, 3.48676679721458799471014036754, 4.67721602825181243574222086979, 5.91316601444185776173671017552, 6.74059370702176364455561503320, 7.79816672452001124238935212130, 8.501334482097550432416695330238, 9.348397151508070995894079109513, 10.33263660974649146035057332871

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