↑

Properties

Label 2-450-45.32-c1-0-8
Degree $2$
Conductor $450$
Sign $0.597 - 0.801i$
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  + (0.965 + 0.258i)2-s + (1.67 + 0.448i)3-s + (0.866 + 0.499i)4-s + (1.50 + 0.866i)6-s + (−0.896 + 3.34i)7-s + (0.707 + 0.707i)8-s + (2.59 + 1.50i)9-s + (−1.5 + 0.866i)11-s + (1.22 + 1.22i)12-s + (−0.896 − 3.34i)13-s + (−1.73 + 3.00i)14-s + (0.500 + 0.866i)16-s + (−2.12 + 2.12i)17-s + (2.12 + 2.12i)18-s − 7i·19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.965 + 0.258i)3-s + (0.433 + 0.249i)4-s + (0.612 + 0.353i)6-s + (−0.338 + 1.26i)7-s + (0.249 + 0.249i)8-s + (0.866 + 0.5i)9-s + (−0.452 + 0.261i)11-s + (0.353 + 0.353i)12-s + (−0.248 − 0.928i)13-s + (−0.462 + 0.801i)14-s + (0.125 + 0.216i)16-s + (−0.514 + 0.514i)17-s + (0.499 + 0.499i)18-s − 1.60i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.597 - 0.801i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.597 - 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38996 + 1.19895i\)
\(L(\frac12)\) \(\approx\) \(2.38996 + 1.19895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (-1.67 - 0.448i)T \)
5 \( 1 \)
good7 \( 1 + (0.896 - 3.34i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.896 + 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.12 - 2.12i)T - 17iT^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 + (-5.79 + 1.55i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.73 + 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + (10.5 + 6.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 1.34i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.67 - 0.448i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (6.12 - 6.12i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.10 - 11.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + (-2.24 + 8.36i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.29733773819022117640519759403, −10.27444220143920286856113008295, −9.264573733620606485832874253722, −8.537358556287148817477884246608, −7.60046594873248482819708337602, −6.52432228335509431545009415262, −5.34490879216285027426894393499, −4.47151969675397148136500124785, −2.98281538743725725516000433511, −2.42620907496419391841846690218, 1.50536184200965566614119064797, 3.02359974528113291077145738441, 3.88487485221578646724998048494, 4.90123600954574312613076232472, 6.51954447181233661241970694947, 7.16250919923969606689714144764, 8.071174210913598759431251772926, 9.260468705738382572338220288147, 10.10299113013241128124721913456, 10.92262641337149693038101285903

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