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Properties

Label 2-450-25.9-c1-0-4
Degree $2$
Conductor $450$
Sign $0.958 + 0.286i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)2-s + (−0.309 − 0.951i)4-s + (2.17 + 0.530i)5-s + 2.72i·7-s + (−0.951 − 0.309i)8-s + (1.70 − 1.44i)10-s + (4.52 + 3.29i)11-s + (0.282 + 0.388i)13-s + (2.20 + 1.60i)14-s + (−0.809 + 0.587i)16-s + (−3.71 − 1.20i)17-s + (1.27 − 3.93i)19-s + (−0.166 − 2.22i)20-s + (5.32 − 1.72i)22-s + (0.0930 − 0.128i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.154 − 0.475i)4-s + (0.971 + 0.237i)5-s + 1.03i·7-s + (−0.336 − 0.109i)8-s + (0.539 − 0.457i)10-s + (1.36 + 0.992i)11-s + (0.0782 + 0.107i)13-s + (0.589 + 0.428i)14-s + (−0.202 + 0.146i)16-s + (−0.900 − 0.292i)17-s + (0.293 − 0.902i)19-s + (−0.0373 − 0.498i)20-s + (1.13 − 0.368i)22-s + (0.0194 − 0.0267i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.958 + 0.286i$
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: \((2,\ 450,\ (\ :1/2),\ 0.958 + 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01512 - 0.294601i\)
\(L(\frac12)\) \(\approx\) \(2.01512 - 0.294601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.530i)T \)
good7 \( 1 - 2.72iT - 7T^{2} \)
11 \( 1 + (-4.52 - 3.29i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.282 - 0.388i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.71 + 1.20i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.27 + 3.93i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.0930 + 0.128i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.17 + 6.68i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.133 + 0.410i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.56 + 2.14i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.86 - 4.98i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.49iT - 43T^{2} \)
47 \( 1 + (-6.43 + 2.08i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.31 - 1.72i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.15 + 4.47i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.97 + 4.34i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.96 + 0.638i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.18 - 6.71i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.99 - 11.0i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.95 + 6.02i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.0 + 3.58i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.38 + 1.00i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (11.2 - 3.66i)T + (78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.28498970721603952351371443565, −10.05917194134306525110153447089, −9.342238935380739494665159725322, −8.814995967849326372760755605731, −7.02145580601674799373803114844, −6.27005074523105233686754367205, −5.26334303245294030448163976408, −4.21471931981816256278024760264, −2.67344976533614005362310430609, −1.78123874102214784031766967759, 1.41448181658267477941922708751, 3.37613307584424556915080177567, 4.34598396102670409490877609647, 5.59972320522324412478202593784, 6.40953911241288362705218337798, 7.18002966228882146811840197442, 8.503681663973238244362301904703, 9.142772693756822243457653619321, 10.24958234596130242704438911950, 11.07613909696902320785364005471

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