↑

Properties

Label 2-450-25.9-c1-0-3
Degree $2$
Conductor $450$
Sign $0.0669 - 0.997i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (1.36 + 1.76i)5-s − 0.533i·7-s + (0.951 + 0.309i)8-s + (−2.23 + 0.0655i)10-s + (−1.16 − 0.843i)11-s + (3.86 + 5.31i)13-s + (0.431 + 0.313i)14-s + (−0.809 + 0.587i)16-s + (0.911 + 0.296i)17-s + (−0.0657 + 0.202i)19-s + (1.26 − 1.84i)20-s + (1.36 − 0.443i)22-s + (2.21 − 3.04i)23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (0.611 + 0.791i)5-s − 0.201i·7-s + (0.336 + 0.109i)8-s + (−0.706 + 0.0207i)10-s + (−0.349 − 0.254i)11-s + (1.07 + 1.47i)13-s + (0.115 + 0.0838i)14-s + (−0.202 + 0.146i)16-s + (0.221 + 0.0718i)17-s + (−0.0150 + 0.0464i)19-s + (0.281 − 0.412i)20-s + (0.290 − 0.0944i)22-s + (0.461 − 0.634i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0669 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0669 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0669 - 0.997i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.0669 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895381 + 0.837315i\)
\(L(\frac12)\) \(\approx\) \(0.895381 + 0.837315i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-1.36 - 1.76i)T \)
good7 \( 1 + 0.533iT - 7T^{2} \)
11 \( 1 + (1.16 + 0.843i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.86 - 5.31i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.911 - 0.296i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0657 - 0.202i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.21 + 3.04i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.91 - 5.89i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.722 - 2.22i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.38 + 3.28i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.42 - 4.66i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 + (-9.65 + 3.13i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.07 + 0.999i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.08 - 4.42i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (10.1 + 7.38i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (6.57 + 2.13i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.12 + 9.62i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.21 + 11.3i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.79 + 14.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-15.5 - 5.04i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (4.54 + 3.30i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.29 + 1.71i)T + (78.4 - 57.0i)T^{2} \)
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

−10.87450342405544465457952005656, −10.54740650091274630215424209867, −9.327259886955742744484691380199, −8.728017034549074230887056994906, −7.50107530474650586638966078638, −6.62763277862627791018474936855, −6.00401327094189916538655863120, −4.69913427703650989869780967965, −3.26133425619655612351406325163, −1.65722295604613156540362349940, 0.981222829572638720647023454104, 2.45762407790186231454413407744, 3.79183906573930020240563231477, 5.19344795597351075167362237587, 5.94225882446333600605738546754, 7.47043721898257656986625246121, 8.416627482439743043771620294754, 9.051315545035418288634496949075, 10.11735760424798957058184448217, 10.62279234204921679522533391992

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