↑

Properties

Label 2-600-120.53-c1-0-42
Degree $2$
Conductor $600$
Sign $0.905 - 0.424i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
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  + (1.37 − 0.313i)2-s + (0.378 + 1.69i)3-s + (1.80 − 0.865i)4-s + (1.05 + 2.21i)6-s + (0.699 − 0.699i)7-s + (2.21 − 1.75i)8-s + (−2.71 + 1.27i)9-s + 4.24·11-s + (2.14 + 2.71i)12-s + (−2.08 + 2.08i)13-s + (0.745 − 1.18i)14-s + (2.50 − 3.12i)16-s + (0.0541 + 0.0541i)17-s + (−3.33 + 2.61i)18-s + 4.52·19-s + ⋯
L(s)  = 1  + (0.975 − 0.221i)2-s + (0.218 + 0.975i)3-s + (0.901 − 0.432i)4-s + (0.429 + 0.902i)6-s + (0.264 − 0.264i)7-s + (0.783 − 0.621i)8-s + (−0.904 + 0.426i)9-s + 1.27·11-s + (0.619 + 0.785i)12-s + (−0.577 + 0.577i)13-s + (0.199 − 0.316i)14-s + (0.625 − 0.780i)16-s + (0.0131 + 0.0131i)17-s + (−0.787 + 0.616i)18-s + 1.03·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91582 + 0.650127i\)
\(L(\frac12)\) \(\approx\) \(2.91582 + 0.650127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.313i)T \)
3 \( 1 + (-0.378 - 1.69i)T \)
5 \( 1 \)
good7 \( 1 + (-0.699 + 0.699i)T - 7iT^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 13iT^{2} \)
17 \( 1 + (-0.0541 - 0.0541i)T + 17iT^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 + (3.48 - 3.48i)T - 23iT^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (-7.29 - 7.29i)T + 37iT^{2} \)
41 \( 1 + 8.74iT - 41T^{2} \)
43 \( 1 + (4.60 - 4.60i)T - 43iT^{2} \)
47 \( 1 + (8.05 + 8.05i)T + 47iT^{2} \)
53 \( 1 + (-0.260 - 0.260i)T + 53iT^{2} \)
59 \( 1 + 2.89iT - 59T^{2} \)
61 \( 1 + 6.70iT - 61T^{2} \)
67 \( 1 + (1.75 + 1.75i)T + 67iT^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + (-4.53 - 4.53i)T + 73iT^{2} \)
79 \( 1 - 1.46iT - 79T^{2} \)
83 \( 1 + (5.61 + 5.61i)T + 83iT^{2} \)
89 \( 1 - 5.42T + 89T^{2} \)
97 \( 1 + (11.3 - 11.3i)T - 97iT^{2} \)
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

−10.96379672260856718146734959335, −9.747672414155611610638886494592, −9.448906791173317362503743761507, −8.006833883356258688394657154421, −6.98890291801992463185390598877, −5.90271047846272031387583767340, −4.95010162829394168130469160550, −4.07309767250789654831398674253, −3.32951743116358537076844743234, −1.83077874229435811996983355223, 1.56067685833282759433287177117, 2.79360147155128132214105350474, 3.88091965512539289292770784577, 5.23250475846868372195864551655, 6.05815990728073355007290348802, 6.97488532575349822807814971563, 7.66302720505510704557826097674, 8.583983468091798370977423960277, 9.645307091717772621849368486632, 11.08123331477218600545716454442

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