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Properties

Label 2-450-25.19-c1-0-2
Degree $2$
Conductor $450$
Sign $0.938 - 0.344i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (−1.97 − 1.03i)5-s + 0.329i·7-s + (−0.587 + 0.809i)8-s + (2.20 + 0.377i)10-s + (1.55 + 4.78i)11-s + (−0.458 − 0.148i)13-s + (−0.101 − 0.313i)14-s + (0.309 − 0.951i)16-s + (3.98 − 5.49i)17-s + (4.40 + 3.19i)19-s + (−2.21 + 0.322i)20-s + (−2.95 − 4.07i)22-s + (6.18 − 2.00i)23-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.404 − 0.293i)4-s + (−0.885 − 0.465i)5-s + 0.124i·7-s + (−0.207 + 0.286i)8-s + (0.696 + 0.119i)10-s + (0.468 + 1.44i)11-s + (−0.127 − 0.0413i)13-s + (−0.0271 − 0.0837i)14-s + (0.0772 − 0.237i)16-s + (0.967 − 1.33i)17-s + (1.00 + 0.733i)19-s + (−0.494 + 0.0720i)20-s + (−0.630 − 0.868i)22-s + (1.28 − 0.418i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.938 - 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907987 + 0.161547i\)
\(L(\frac12)\) \(\approx\) \(0.907987 + 0.161547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 \)
5 \( 1 + (1.97 + 1.03i)T \)
good7 \( 1 - 0.329iT - 7T^{2} \)
11 \( 1 + (-1.55 - 4.78i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.458 + 0.148i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.98 + 5.49i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.40 - 3.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.18 + 2.00i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.87 + 3.53i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.06 + 0.775i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.741 + 0.241i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.86 - 11.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 + (-2.57 - 3.54i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.990 + 1.36i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.313 + 0.966i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.29 - 3.98i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.84 - 2.54i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.62 + 3.35i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.79 - 0.909i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.86 + 4.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (10.1 - 13.9i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.06 - 3.28i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.58 + 7.69i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.20819971363961889324229871895, −9.905082068592387490951644649858, −9.457486339317669390507583118204, −8.355766674600982922089227505071, −7.48270780194870265446794145521, −6.91220904649611697166056256191, −5.36237562906596309003260032035, −4.46075891808683625890948482143, −2.96186739093583104329438331043, −1.13016216534445830905088702022, 0.977468812529455156453294713039, 3.05462259299309576793852231805, 3.74581141280329981360722711762, 5.41579080828766922965541747950, 6.65955977971642890586709399015, 7.47058496732039577602159899453, 8.423895162213995441879519566370, 9.064388776047908751424905053854, 10.37883015316896330583954125881, 10.93691822017655764555923499900

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