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Properties

Label 2-450-225.94-c1-0-10
Degree $2$
Conductor $450$
Sign $-0.382 - 0.923i$
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  + (0.743 + 0.669i)2-s + (1.71 + 0.239i)3-s + (0.104 + 0.994i)4-s + (−1.65 + 1.50i)5-s + (1.11 + 1.32i)6-s + (−2.80 + 1.61i)7-s + (−0.587 + 0.809i)8-s + (2.88 + 0.821i)9-s + (−2.23 + 0.0157i)10-s + (−1.71 + 1.90i)11-s + (−0.0589 + 1.73i)12-s + (−1.49 + 1.34i)13-s + (−3.16 − 0.672i)14-s + (−3.19 + 2.19i)15-s + (−0.978 + 0.207i)16-s + (4.08 − 5.61i)17-s + ⋯
L(s)  = 1  + (0.525 + 0.473i)2-s + (0.990 + 0.138i)3-s + (0.0522 + 0.497i)4-s + (−0.738 + 0.674i)5-s + (0.454 + 0.541i)6-s + (−1.05 + 0.611i)7-s + (−0.207 + 0.286i)8-s + (0.961 + 0.273i)9-s + (−0.707 + 0.00497i)10-s + (−0.517 + 0.575i)11-s + (−0.0170 + 0.499i)12-s + (−0.414 + 0.372i)13-s + (−0.845 − 0.179i)14-s + (−0.824 + 0.565i)15-s + (−0.244 + 0.0519i)16-s + (0.989 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09544 + 1.63959i\)
\(L(\frac12)\) \(\approx\) \(1.09544 + 1.63959i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.743 - 0.669i)T \)
3 \( 1 + (-1.71 - 0.239i)T \)
5 \( 1 + (1.65 - 1.50i)T \)
good7 \( 1 + (2.80 - 1.61i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.71 - 1.90i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (1.49 - 1.34i)T + (1.35 - 12.9i)T^{2} \)
17 \( 1 + (-4.08 + 5.61i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.00 - 4.36i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.496 + 2.33i)T + (-21.0 - 9.35i)T^{2} \)
29 \( 1 + (-1.79 - 0.797i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (-0.0214 + 0.00955i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-6.63 - 2.15i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.50 + 2.77i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (-4.51 + 2.60i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.96 + 4.41i)T + (-31.4 - 34.9i)T^{2} \)
53 \( 1 + (2.21 + 3.04i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.38 + 1.54i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (3.28 - 3.65i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (-5.92 - 13.3i)T + (-44.8 + 49.7i)T^{2} \)
71 \( 1 + (7.76 - 5.63i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (10.5 - 3.44i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.1 + 4.97i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (2.96 + 0.311i)T + (81.1 + 17.2i)T^{2} \)
89 \( 1 + (-0.232 - 0.716i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.49 + 12.3i)T + (-64.9 - 72.0i)T^{2} \)
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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

−11.71745966689330432073932659947, −10.11822014219740299086796949520, −9.654347451180384136649773030699, −8.499515228662547131607944034072, −7.41167777724121587916210476898, −7.12280634221984451873212683462, −5.70030053085450377111435361131, −4.45587127307544623538508417921, −3.24754426409950786322129197288, −2.70484196575978506782233512982, 0.997556759147545562434243211342, 2.99955154640242977227215216893, 3.57024113484716454700665116383, 4.69845912390393326360800210733, 5.99916713504128675920534905519, 7.37608941665310305607360621800, 7.965210454294448114000110366449, 9.164888437457075894051117308667, 9.853202192245000727833334471070, 10.77540862306695592623413627893

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