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Properties

Label 2-450-225.92-c1-0-3
Degree $2$
Conductor $450$
Sign $-0.374 - 0.927i$
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  + (0.838 − 0.544i)2-s + (−1.68 + 0.412i)3-s + (0.406 − 0.913i)4-s + (1.37 + 1.76i)5-s + (−1.18 + 1.26i)6-s + (−4.81 + 1.29i)7-s + (−0.156 − 0.987i)8-s + (2.65 − 1.38i)9-s + (2.11 + 0.731i)10-s + (−0.151 + 0.713i)11-s + (−0.306 + 1.70i)12-s + (−2.26 + 3.49i)13-s + (−3.33 + 3.70i)14-s + (−3.03 − 2.39i)15-s + (−0.669 − 0.743i)16-s + (−4.18 + 0.662i)17-s + ⋯
L(s)  = 1  + (0.593 − 0.385i)2-s + (−0.971 + 0.238i)3-s + (0.203 − 0.456i)4-s + (0.614 + 0.788i)5-s + (−0.484 + 0.515i)6-s + (−1.82 + 0.487i)7-s + (−0.0553 − 0.349i)8-s + (0.886 − 0.463i)9-s + (0.668 + 0.231i)10-s + (−0.0457 + 0.215i)11-s + (−0.0885 + 0.492i)12-s + (−0.629 + 0.968i)13-s + (−0.892 + 0.990i)14-s + (−0.784 − 0.619i)15-s + (−0.167 − 0.185i)16-s + (−1.01 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433122 + 0.642324i\)
\(L(\frac12)\) \(\approx\) \(0.433122 + 0.642324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.838 + 0.544i)T \)
3 \( 1 + (1.68 - 0.412i)T \)
5 \( 1 + (-1.37 - 1.76i)T \)
good7 \( 1 + (4.81 - 1.29i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.151 - 0.713i)T + (-10.0 - 4.47i)T^{2} \)
13 \( 1 + (2.26 - 3.49i)T + (-5.28 - 11.8i)T^{2} \)
17 \( 1 + (4.18 - 0.662i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.62 - 2.23i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.15 + 0.0603i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (-0.733 - 6.97i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.418 + 3.98i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-7.34 + 3.74i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-0.220 - 1.03i)T + (-37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.27 - 8.48i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.96 + 5.63i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (-3.53 - 0.559i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (10.7 - 2.29i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (7.83 + 1.66i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-9.50 + 7.69i)T + (13.9 - 65.5i)T^{2} \)
71 \( 1 + (-1.09 - 1.50i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.801 - 0.408i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-5.25 + 0.551i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-7.88 - 3.02i)T + (61.6 + 55.5i)T^{2} \)
89 \( 1 + (-2.03 - 6.24i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-9.70 + 11.9i)T + (-20.1 - 94.8i)T^{2} \)
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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

−11.31066222461958118863946559224, −10.59821407172059960784164878921, −9.644395796969174575647869431800, −9.353567444762428975149539099266, −7.06336301764998852850199272114, −6.44286930956895000512014871102, −5.90901934770760695771029153630, −4.61527074854222988523359401334, −3.41971533856268844235168891502, −2.21304938663673057212599319303, 0.41773331345035272559360266305, 2.67743314828998508699712393727, 4.21608475761449588342484786897, 5.18933638341179782391132858142, 6.19301233495970658636556927561, 6.64620982896165080632037618127, 7.77223546768643501967912533469, 9.171191548433766786460109888955, 9.984001163297026128584453096384, 10.78709625085766711468366510066

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