↑

Properties

Label 2-600-120.59-c1-0-43
Degree $2$
Conductor $600$
Sign $0.940 - 0.338i$
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  + (−0.199 + 1.40i)2-s + (1.65 − 0.520i)3-s + (−1.92 − 0.557i)4-s + (0.400 + 2.41i)6-s − 1.92·7-s + (1.16 − 2.57i)8-s + (2.45 − 1.72i)9-s − 4.02i·11-s + (−3.46 + 0.0792i)12-s + 4.81·13-s + (0.383 − 2.69i)14-s + (3.37 + 2.14i)16-s + 5.23·17-s + (1.91 + 3.78i)18-s + 0.684·19-s + ⋯
L(s)  = 1  + (−0.140 + 0.990i)2-s + (0.953 − 0.300i)3-s + (−0.960 − 0.278i)4-s + (0.163 + 0.986i)6-s − 0.728·7-s + (0.411 − 0.911i)8-s + (0.819 − 0.573i)9-s − 1.21i·11-s + (−0.999 + 0.0228i)12-s + 1.33·13-s + (0.102 − 0.721i)14-s + (0.844 + 0.535i)16-s + 1.26·17-s + (0.452 + 0.891i)18-s + 0.157·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.940 - 0.338i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.940 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69303 + 0.295249i\)
\(L(\frac12)\) \(\approx\) \(1.69303 + 0.295249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.199 - 1.40i)T \)
3 \( 1 + (-1.65 + 0.520i)T \)
5 \( 1 \)
good7 \( 1 + 1.92T + 7T^{2} \)
11 \( 1 + 4.02iT - 11T^{2} \)
13 \( 1 - 4.81T + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 0.684T + 19T^{2} \)
23 \( 1 - 1.72iT - 23T^{2} \)
29 \( 1 + 6.99T + 29T^{2} \)
31 \( 1 - 4.23iT - 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 - 1.04iT - 43T^{2} \)
47 \( 1 + 7.55iT - 47T^{2} \)
53 \( 1 + 4.08iT - 53T^{2} \)
59 \( 1 + 0.994iT - 59T^{2} \)
61 \( 1 + 3.16iT - 61T^{2} \)
67 \( 1 - 14.8iT - 67T^{2} \)
71 \( 1 + 9.28T + 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 9.25iT - 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 - 0.829iT - 89T^{2} \)
97 \( 1 - 1.45iT - 97T^{2} \)
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

−10.34798211332928837720804957670, −9.518733059520506433944764257303, −8.760570701093140651985772674187, −8.117249074692809903647931361586, −7.25308465276905569283402550378, −6.25776527942973021462874105600, −5.57100099045956269434697818799, −3.84694499520824665099072499721, −3.26392495848521118586000008366, −1.10248650905884701525080644304, 1.51865017370737165320144446033, 2.81971437793366528604785632520, 3.70509493046034934522670995765, 4.54907078820580139504036105521, 5.94844631943791405285608035263, 7.46635290197220453067408952676, 8.129890484038190991726236115415, 9.307016029183350177944675664286, 9.608609614948084981226017594433, 10.44660458643260041468589701025

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