Properties

Label 2-600-24.11-c1-0-14
Degree $2$
Conductor $600$
Sign $-0.333 - 0.942i$
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 + (−1.55 − 1.89i)6-s + 4.34i·7-s + (−2.45 − 1.40i)8-s + (0.448 + 2.96i)9-s − 1.83i·11-s + (1.70 + 3.01i)12-s − 0.588i·13-s + (1.06 − 6.05i)14-s + (3.07 + 2.55i)16-s + 5.37i·17-s + (0.0995 − 4.24i)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 + (−0.634 − 0.773i)6-s + 1.64i·7-s + (−0.867 − 0.497i)8-s + (0.149 + 0.988i)9-s − 0.553i·11-s + (0.491 + 0.871i)12-s − 0.163i·13-s + (0.283 − 1.61i)14-s + (0.768 + 0.639i)16-s + 1.30i·17-s + (0.0234 − 0.999i)18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.333 - 0.942i$
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: \((2,\ 600,\ (\ :1/2),\ -0.333 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620919 + 0.878159i\)
\(L(\frac12)\) \(\approx\) \(0.620919 + 0.878159i\)
\(L(\frac{3}{2})\) 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

−10.67223029375667435504711543002, −9.950043561079335930411184396890, −9.042297907055169243873239774004, −8.431854442529069641250498586429, −8.029357277469881746348978385565, −6.41969460415730678202857187050, −5.65227145305005123363305185428, −4.08060925097914978171142612694, −2.82559893413806442040464556469, −2.04092839488671702835447117473, 0.73619685926379585741725399922, 2.05637986955372601787924766757, 3.37969882414569188846773212722, 4.71278936704390545864826749965, 6.64210883953793741264534426517, 6.86013889989203281067262873873, 7.81796730071062812125393037423, 8.473888918737583832263418511646, 9.551795697182450899377244938636, 10.16651622349517928862842832731

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