↑

Properties

Label 2-450-5.4-c1-0-5
Degree $2$
Conductor $450$
Sign $0.894 + 0.447i$
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  + i·2-s − 4-s − 4i·7-s − i·8-s − 2i·13-s + 4·14-s + 16-s − 6i·17-s + 4·19-s + 2·26-s + 4i·28-s − 6·29-s + 8·31-s + i·32-s + 6·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.51i·7-s − 0.353i·8-s − 0.554i·13-s + 1.06·14-s + 0.250·16-s − 1.45i·17-s + 0.917·19-s + 0.392·26-s + 0.755i·28-s − 1.11·29-s + 1.43·31-s + 0.176i·32-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19121 - 0.281208i\)
\(L(\frac12)\) \(\approx\) \(1.19121 - 0.281208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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

−10.92364972653550439038892589587, −9.981668100371647146233798173812, −9.305081061958847708933319532503, −7.921454892623089513938599914123, −7.42310713674311558565678035377, −6.53514944981278129352164932106, −5.27976105288632726974539227014, −4.34799478233135075827123058536, −3.14981777812299502186399256676, −0.821288108852431930051466738368, 1.76826063882083140168211212390, 2.90916387940588836638728492533, 4.19385360768860040961610482192, 5.45315140469871906146730803665, 6.21704172593643872004407751652, 7.72399626762213868515367902971, 8.734518401528348357812219131052, 9.310092617995571481528373302408, 10.29924749053468720653084671664, 11.30299090140795773444738240252

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