Properties

Label 2-600-24.11-c1-0-63
Degree $2$
Conductor $600$
Sign $0.982 + 0.188i$
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  + (1.39 − 0.244i)2-s + (1.31 + 1.12i)3-s + (1.88 − 0.680i)4-s + (2.10 + 1.25i)6-s − 4.34i·7-s + (2.45 − 1.40i)8-s + (0.448 + 2.96i)9-s − 1.83i·11-s + (3.23 + 1.23i)12-s + 0.588i·13-s + (−1.06 − 6.05i)14-s + (3.07 − 2.55i)16-s + 5.37i·17-s + (1.34 + 4.02i)18-s − 5.38·19-s + ⋯
L(s)  = 1  + (0.984 − 0.172i)2-s + (0.758 + 0.652i)3-s + (0.940 − 0.340i)4-s + (0.859 + 0.511i)6-s − 1.64i·7-s + (0.867 − 0.497i)8-s + (0.149 + 0.988i)9-s − 0.553i·11-s + (0.934 + 0.355i)12-s + 0.163i·13-s + (−0.283 − 1.61i)14-s + (0.768 − 0.639i)16-s + 1.30i·17-s + (0.317 + 0.948i)18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.36383 - 0.320256i\)
\(L(\frac12)\) \(\approx\) \(3.36383 - 0.320256i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.244i)T \)
3 \( 1 + (-1.31 - 1.12i)T \)
5 \( 1 \)
good7 \( 1 + 4.34iT - 7T^{2} \)
11 \( 1 + 1.83iT - 11T^{2} \)
13 \( 1 - 0.588iT - 13T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 - 7.06iT - 31T^{2} \)
37 \( 1 - 2.72iT - 37T^{2} \)
41 \( 1 - 3.42iT - 41T^{2} \)
43 \( 1 - 2.96T + 43T^{2} \)
47 \( 1 - 9.81T + 47T^{2} \)
53 \( 1 + 6.65T + 53T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + 9.27iT - 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 4.42T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 4.21iT - 89T^{2} \)
97 \( 1 - 2.16T + 97T^{2} \)
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

−10.66410005853290377620329017757, −10.18036161684282697515977013315, −8.903770126163541897551009135787, −7.86024021617199678527803501434, −7.04729940403364792745155071110, −5.97696505837457167843017454082, −4.63946439858543564947066138709, −3.98752917807885794014094525718, −3.22641115944478796538326408397, −1.67432200816642259972644948148, 2.12656947822912315617696853113, 2.66065144569346782930692920421, 3.99089304078704287993565475874, 5.27317228039981043267767662425, 6.08630056060033112257853977728, 7.06555341140885550059330675974, 7.86486731583879888271772484637, 8.870930134151042666121438246521, 9.545390019377674662568650000114, 11.07137099176468788909632362818

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