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Properties

Label 2-450-25.9-c1-0-1
Degree $2$
Conductor $450$
Sign $0.511 - 0.859i$
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 + (−2.03 + 0.917i)5-s + 4.80i·7-s + (−0.951 − 0.309i)8-s + (−0.456 + 2.18i)10-s + (−0.714 − 0.518i)11-s + (1.66 + 2.28i)13-s + (3.88 + 2.82i)14-s + (−0.809 + 0.587i)16-s + (1.57 + 0.512i)17-s + (−1.66 + 5.11i)19-s + (1.50 + 1.65i)20-s + (−0.839 + 0.272i)22-s + (−3.44 + 4.73i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.912 + 0.410i)5-s + 1.81i·7-s + (−0.336 − 0.109i)8-s + (−0.144 + 0.692i)10-s + (−0.215 − 0.156i)11-s + (0.460 + 0.633i)13-s + (1.03 + 0.755i)14-s + (−0.202 + 0.146i)16-s + (0.382 + 0.124i)17-s + (−0.381 + 1.17i)19-s + (0.335 + 0.370i)20-s + (−0.178 + 0.0581i)22-s + (−0.717 + 0.987i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.511 - 0.859i$
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: \((2,\ 450,\ (\ :1/2),\ 0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04849 + 0.595905i\)
\(L(\frac12)\) \(\approx\) \(1.04849 + 0.595905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 - 0.917i)T \)
good7 \( 1 - 4.80iT - 7T^{2} \)
11 \( 1 + (0.714 + 0.518i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.66 - 2.28i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.57 - 0.512i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.66 - 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.44 - 4.73i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.10 - 3.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.22 + 9.93i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.02 + 1.41i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.40 - 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.27iT - 43T^{2} \)
47 \( 1 + (-8.29 + 2.69i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.37 + 1.09i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.37 - 6.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.0697 - 0.0506i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (11.3 + 3.69i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.08 - 3.33i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.24 + 5.84i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.88 - 11.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-12.6 - 4.10i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (15.1 + 10.9i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-17.3 + 5.64i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.56698038224173308165890734360, −10.54090261011492797482712638239, −9.476639556652904304590343606828, −8.577901207506663808047974874212, −7.75045258894515169868332397077, −6.23944450708875103843513537895, −5.59735363101603421210941587021, −4.22748663340421177858681206488, −3.20128960919686035427717725554, −2.02760255949995575913088898211, 0.67314002840308064039861946339, 3.25020921686851510373502221027, 4.24789101580869080203328740102, 4.90624133806625264291805474137, 6.45582361024823263163598823352, 7.29795645752967332187105685513, 7.965442146033530783946749539242, 8.832900649105794751437854596112, 10.30313511615505191770176911965, 10.83257846285870753026971360136

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