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Properties

Label 2-450-225.106-c1-0-12
Degree $2$
Conductor $450$
Sign $0.651 + 0.759i$
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  + (−0.978 − 0.207i)2-s + (−0.590 − 1.62i)3-s + (0.913 + 0.406i)4-s + (0.712 + 2.11i)5-s + (0.239 + 1.71i)6-s + (1.08 − 1.88i)7-s + (−0.809 − 0.587i)8-s + (−2.30 + 1.92i)9-s + (−0.256 − 2.22i)10-s + (−0.452 − 0.0960i)11-s + (0.122 − 1.72i)12-s + (4.54 − 0.966i)13-s + (−1.45 + 1.61i)14-s + (3.03 − 2.41i)15-s + (0.669 + 0.743i)16-s + (4.57 + 3.32i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.147i)2-s + (−0.340 − 0.940i)3-s + (0.456 + 0.203i)4-s + (0.318 + 0.947i)5-s + (0.0975 + 0.700i)6-s + (0.410 − 0.711i)7-s + (−0.286 − 0.207i)8-s + (−0.767 + 0.641i)9-s + (−0.0809 − 0.702i)10-s + (−0.136 − 0.0289i)11-s + (0.0354 − 0.498i)12-s + (1.26 − 0.267i)13-s + (−0.388 + 0.431i)14-s + (0.782 − 0.622i)15-s + (0.167 + 0.185i)16-s + (1.11 + 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.651 + 0.759i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.651 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939159 - 0.431792i\)
\(L(\frac12)\) \(\approx\) \(0.939159 - 0.431792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.978 + 0.207i)T \)
3 \( 1 + (0.590 + 1.62i)T \)
5 \( 1 + (-0.712 - 2.11i)T \)
good7 \( 1 + (-1.08 + 1.88i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.452 + 0.0960i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-4.54 + 0.966i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-4.57 - 3.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.634 + 0.460i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.12 + 2.35i)T + (-2.40 - 22.8i)T^{2} \)
29 \( 1 + (0.100 + 0.952i)T + (-28.3 + 6.02i)T^{2} \)
31 \( 1 + (-0.880 + 8.37i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (-2.18 + 6.72i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-6.00 + 1.27i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-3.51 + 6.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.249 + 2.37i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (7.13 - 5.18i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (7.79 - 1.65i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + (-5.79 - 1.23i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (0.672 - 6.39i)T + (-65.5 - 13.9i)T^{2} \)
71 \( 1 + (4.09 - 2.97i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.75 + 14.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.42 - 13.5i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (-10.6 + 4.75i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-2.91 - 8.98i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.36 - 13.0i)T + (-94.8 + 20.1i)T^{2} \)
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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

−10.76147396781178307104495480889, −10.50641675340153192758703316155, −9.116913817966685669873370749463, −7.907115351649159297167425087892, −7.52543252821542301401369692682, −6.37035446875510893786737523460, −5.75954422893376469310065040450, −3.78653598502971202750522614268, −2.41860976723534072409783491155, −1.07295147028982396963816080404, 1.28117604115803339263049925798, 3.16299511533345230470982359564, 4.70288233443628397224257083050, 5.49352427688701050213402148379, 6.32293156158844884076271496691, 7.935206330407062530105681522312, 8.748445752844093097447152285338, 9.320018944592265245704426702620, 10.12978777143050570294461858294, 11.16016313631474661601505951565

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