Properties

Label 2-450-25.9-c1-0-6
Degree $2$
Conductor $450$
Sign $0.0238 + 0.999i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−1.70 + 1.44i)5-s − 1.72i·7-s + (0.951 + 0.309i)8-s + (−0.166 − 2.22i)10-s + (−1.97 − 1.43i)11-s + (−1.74 − 2.40i)13-s + (1.39 + 1.01i)14-s + (−0.809 + 0.587i)16-s + (−3.18 − 1.03i)17-s + (−0.694 + 2.13i)19-s + (1.90 + 1.17i)20-s + (2.32 − 0.754i)22-s + (5.35 − 7.37i)23-s + ⋯
L(s)  = 1  + (−0.415 + 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.762 + 0.646i)5-s − 0.652i·7-s + (0.336 + 0.109i)8-s + (−0.0528 − 0.705i)10-s + (−0.595 − 0.432i)11-s + (−0.484 − 0.666i)13-s + (0.373 + 0.271i)14-s + (−0.202 + 0.146i)16-s + (−0.771 − 0.250i)17-s + (−0.159 + 0.490i)19-s + (0.425 + 0.262i)20-s + (0.495 − 0.160i)22-s + (1.11 − 1.53i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0238 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0238 + 0.999i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.0238 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322464 - 0.314872i\)
\(L(\frac12)\) \(\approx\) \(0.322464 - 0.314872i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (1.70 - 1.44i)T \)
good7 \( 1 + 1.72iT - 7T^{2} \)
11 \( 1 + (1.97 + 1.43i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.74 + 2.40i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.18 + 1.03i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.694 - 2.13i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.35 + 7.37i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.89 + 8.91i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.89 - 5.82i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.94 + 2.67i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.95 - 3.60i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.06iT - 43T^{2} \)
47 \( 1 + (4.74 - 1.54i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.51 + 2.11i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.147 + 0.107i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.16 + 3.02i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (11.3 + 3.67i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.51 - 10.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.95 - 5.44i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.50 - 10.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (8.69 + 2.82i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-6.56 - 4.77i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-9.40 + 3.05i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.61633871726821414508258211384, −10.22320430632109730418203042075, −8.821855263758046801606968882384, −8.024368786177645338138518271207, −7.23405979438107287092121120468, −6.48653498190210351055501452538, −5.17483057228628655282287134298, −4.03952480245351232186578415172, −2.68556387787147667823323273572, −0.31483236115905909042509069562, 1.77404131112848282278957877911, 3.19036534963291407526264812365, 4.49211498639220447791755919303, 5.33767691406201586313716034302, 7.00989664272243381973078910787, 7.75466442099991702351294029699, 9.001093102882277807445514219576, 9.159939176919641468350893826007, 10.52226598141174595520318677858, 11.43145383259650918140393620230

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