Properties

Label 2-450-45.23-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.300 - 0.953i$
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  + (0.258 − 0.965i)2-s + (−1.67 + 0.448i)3-s + (−0.866 − 0.499i)4-s + 1.73i·6-s + (−0.707 + 0.707i)8-s + (2.59 − 1.50i)9-s + (−1.5 + 0.866i)11-s + (1.67 + 0.448i)12-s + (−6.69 + 1.79i)13-s + (0.500 + 0.866i)16-s + (−2.12 − 2.12i)17-s + (−0.776 − 2.89i)18-s + 4i·19-s + (0.448 + 1.67i)22-s + (1.55 + 5.79i)23-s + (0.866 − 1.5i)24-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.965 + 0.258i)3-s + (−0.433 − 0.249i)4-s + 0.707i·6-s + (−0.249 + 0.249i)8-s + (0.866 − 0.5i)9-s + (−0.452 + 0.261i)11-s + (0.482 + 0.129i)12-s + (−1.85 + 0.497i)13-s + (0.125 + 0.216i)16-s + (−0.514 − 0.514i)17-s + (−0.183 − 0.683i)18-s + 0.917i·19-s + (0.0955 + 0.356i)22-s + (0.323 + 1.20i)23-s + (0.176 − 0.306i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.180243 + 0.245884i\)
\(L(\frac12)\) \(\approx\) \(0.180243 + 0.245884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
5 \( 1 \)
good7 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.69 - 1.79i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.12 + 2.12i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-1.55 - 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.73 - 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 8.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.10 + 11.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.896 + 3.34i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (4.89 + 4.89i)T + 73iT^{2} \)
79 \( 1 + (3.46 - 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.69 - 2.32i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (-15.0 - 4.03i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.58676896728821843192415473383, −10.35830406397746661363402169428, −9.967614058129938504593253002180, −9.014035957040260557167061017312, −7.51098644418504124044281764924, −6.67554249959978129819931037939, −5.21487210689618369565076716029, −4.84996107510514202045537497253, −3.45065169928727962165776898290, −1.86935696868450519155710332313, 0.19224318159838953662073708944, 2.53896050480412485614984564258, 4.44111732952015781981362646042, 5.13025927054656271934021409533, 6.11703638023450213492472830376, 7.07133574963313552227755613380, 7.73048614041314370757164880935, 8.918346867630928656569936062450, 10.07390505741066109443630102132, 10.77335001856293753803851687013

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