↑

Properties

Label 2-30e2-45.29-c2-0-27
Degree $2$
Conductor $900$
Sign $0.751 + 0.660i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.56 − 1.54i)3-s + (1.25 + 0.725i)7-s + (4.20 − 7.95i)9-s + (8.07 + 4.66i)11-s + (9.70 − 5.60i)13-s + 4.16·17-s − 3.14·19-s + (4.34 − 0.0803i)21-s + (6.69 + 11.5i)23-s + (−1.49 − 26.9i)27-s + (−12.6 − 7.33i)29-s + (7.07 + 12.2i)31-s + (27.9 − 0.516i)33-s − 18.0i·37-s + (16.2 − 29.4i)39-s + ⋯
L(s)  = 1  + (0.856 − 0.515i)3-s + (0.179 + 0.103i)7-s + (0.467 − 0.883i)9-s + (0.733 + 0.423i)11-s + (0.746 − 0.430i)13-s + 0.244·17-s − 0.165·19-s + (0.207 − 0.00382i)21-s + (0.291 + 0.504i)23-s + (−0.0553 − 0.998i)27-s + (−0.437 − 0.252i)29-s + (0.228 + 0.395i)31-s + (0.847 − 0.0156i)33-s − 0.487i·37-s + (0.417 − 0.754i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.751 + 0.660i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.751 + 0.660i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.917913462\)
\(L(\frac12)\) \(\approx\) \(2.917913462\)
\(L(2)\) 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 \( 1 + (-2.56 + 1.54i)T \)
5 \( 1 \)
good7 \( 1 + (-1.25 - 0.725i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.07 - 4.66i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.70 + 5.60i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 4.16T + 289T^{2} \)
19 \( 1 + 3.14T + 361T^{2} \)
23 \( 1 + (-6.69 - 11.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (12.6 + 7.33i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-7.07 - 12.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 18.0iT - 1.36e3T^{2} \)
41 \( 1 + (-17.6 + 10.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-28.9 - 16.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (0.946 - 1.63i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 97.8T + 2.80e3T^{2} \)
59 \( 1 + (-28.9 + 16.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-29.6 + 51.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (83.0 - 47.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 97.3iT - 5.04e3T^{2} \)
73 \( 1 - 90.1iT - 5.32e3T^{2} \)
79 \( 1 + (-66.7 + 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-3.74 + 6.48i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (56.1 + 32.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−9.598444294225614233298581537635, −8.933188802482542254034878617547, −8.163292870543814899437969858936, −7.34754545687090523591466034995, −6.54150939550046573252432054256, −5.55053080930000075156923082896, −4.18248388659633897480949177278, −3.36614172981262562939560012084, −2.14216116061995893978890408541, −1.03664115154519265847627395593, 1.28075092930163904683887449970, 2.60366235113368521730854158737, 3.73230193333934626734119715311, 4.38234221117187954358874495520, 5.59897583640868801409537328063, 6.65514738765382291309145311660, 7.61797880526282103231234644738, 8.557830549848981832761429286778, 9.029348412790942590698847580247, 9.910776842386643791108904404693

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