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Properties

Label 2-450-15.8-c1-0-3
Degree $2$
Conductor $450$
Sign $0.662 + 0.749i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (3 − 3i)7-s + (0.707 − 0.707i)8-s + 4.24i·11-s + (3 + 3i)13-s − 4.24·14-s − 1.00·16-s + (−4.24 − 4.24i)17-s − 2i·19-s + (3 − 3i)22-s + (4.24 − 4.24i)23-s − 4.24i·26-s + (3.00 + 3.00i)28-s + 8.48·29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (1.13 − 1.13i)7-s + (0.250 − 0.250i)8-s + 1.27i·11-s + (0.832 + 0.832i)13-s − 1.13·14-s − 0.250·16-s + (−1.02 − 1.02i)17-s − 0.458i·19-s + (0.639 − 0.639i)22-s + (0.884 − 0.884i)23-s − 0.832i·26-s + (0.566 + 0.566i)28-s + 1.57·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13459 - 0.511656i\)
\(L(\frac12)\) \(\approx\) \(1.13459 - 0.511656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (4.24 - 4.24i)T - 53iT^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + (6 - 6i)T - 67iT^{2} \)
71 \( 1 - 8.48iT - 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 - 8iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 + (12 - 12i)T - 97iT^{2} \)
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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

−10.97497678768499833050174191972, −10.22225304100640072573837852887, −9.177398932681114643091932314707, −8.384909946225847808031976332846, −7.27378098614353589435576664411, −6.73497115525536461807279059158, −4.66137205900719789171467481818, −4.34312292594236896339646801923, −2.50712236879917508316777111661, −1.16381502625657574254519404517, 1.40599523600476505036494093736, 3.02845897639445743033854465535, 4.70558025091208987450440556086, 5.76811156192630985426305303109, 6.34643028794037614256392179607, 7.977958256632927353916050913093, 8.406998596438304340287373599065, 9.026357908353123323691007293185, 10.41691304458555711885822477616, 11.12484859597084846810382096457

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