↑

Properties

Label 2-600-5.4-c3-0-9
Degree $2$
Conductor $600$
Sign $0.894 - 0.447i$
Analytic cond. $35.4011$
Root an. cond. $5.94988$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 19.8i·7-s − 9·9-s − 70.6·11-s + 0.761i·13-s + 108. i·17-s + 125.·19-s + 59.6·21-s − 40.8i·23-s − 27i·27-s + 140.·29-s + 296.·31-s − 211. i·33-s − 26.8i·37-s − 2.28·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.07i·7-s − 0.333·9-s − 1.93·11-s + 0.0162i·13-s + 1.54i·17-s + 1.51·19-s + 0.619·21-s − 0.370i·23-s − 0.192i·27-s + 0.902·29-s + 1.72·31-s − 1.11i·33-s − 0.119i·37-s − 0.00937·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(35.4011\)
Root analytic conductor: \(5.94988\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.654669540\)
\(L(\frac12)\) \(\approx\) \(1.654669540\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 19.8iT - 343T^{2} \)
11 \( 1 + 70.6T + 1.33e3T^{2} \)
13 \( 1 - 0.761iT - 2.19e3T^{2} \)
17 \( 1 - 108. iT - 4.91e3T^{2} \)
19 \( 1 - 125.T + 6.85e3T^{2} \)
23 \( 1 + 40.8iT - 1.21e4T^{2} \)
29 \( 1 - 140.T + 2.43e4T^{2} \)
31 \( 1 - 296.T + 2.97e4T^{2} \)
37 \( 1 + 26.8iT - 5.06e4T^{2} \)
41 \( 1 + 20.6T + 6.89e4T^{2} \)
43 \( 1 + 32.5iT - 7.95e4T^{2} \)
47 \( 1 - 11.4iT - 1.03e5T^{2} \)
53 \( 1 + 159. iT - 1.48e5T^{2} \)
59 \( 1 - 374.T + 2.05e5T^{2} \)
61 \( 1 - 303.T + 2.26e5T^{2} \)
67 \( 1 - 877. iT - 3.00e5T^{2} \)
71 \( 1 - 1.15e3T + 3.57e5T^{2} \)
73 \( 1 + 120. iT - 3.89e5T^{2} \)
79 \( 1 - 1.24e3T + 4.93e5T^{2} \)
83 \( 1 + 654. iT - 5.71e5T^{2} \)
89 \( 1 + 795.T + 7.04e5T^{2} \)
97 \( 1 - 850. iT - 9.12e5T^{2} \)
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

−10.33055816932099659162606545794, −9.811062892740954573861519111129, −8.346819824508185988484754371100, −7.88177672925453740359316459719, −6.77691897358758887412556558103, −5.57438437826176709778940549805, −4.73350815305930488286876769295, −3.69187132258775436101667716947, −2.59774447268057978674307188425, −0.808580234728446253246473815447, 0.70947491313135965140901911626, 2.46198998057567424958090876230, 2.97438993557653925028583762678, 5.04411413641386862694425833658, 5.40335431180196531797906447688, 6.64519107459104054764165866460, 7.68318177402130697292045679222, 8.230155512943669398711520692437, 9.353399312675787757004362468795, 10.09359320333650547312909622338

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