Properties

Label 2-450-45.32-c1-0-11
Degree $2$
Conductor $450$
Sign $-0.597 + 0.801i$
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  + (−0.965 − 0.258i)2-s + (−1.67 − 0.448i)3-s + (0.866 + 0.499i)4-s + (1.50 + 0.866i)6-s + (0.896 − 3.34i)7-s + (−0.707 − 0.707i)8-s + (2.59 + 1.50i)9-s + (−1.5 + 0.866i)11-s + (−1.22 − 1.22i)12-s + (0.896 + 3.34i)13-s + (−1.73 + 3.00i)14-s + (0.500 + 0.866i)16-s + (2.12 − 2.12i)17-s + (−2.12 − 2.12i)18-s − 7i·19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.965 − 0.258i)3-s + (0.433 + 0.249i)4-s + (0.612 + 0.353i)6-s + (0.338 − 1.26i)7-s + (−0.249 − 0.249i)8-s + (0.866 + 0.5i)9-s + (−0.452 + 0.261i)11-s + (−0.353 − 0.353i)12-s + (0.248 + 0.928i)13-s + (−0.462 + 0.801i)14-s + (0.125 + 0.216i)16-s + (0.514 − 0.514i)17-s + (−0.499 − 0.499i)18-s − 1.60i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.597 + 0.801i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.597 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247023 - 0.492408i\)
\(L(\frac12)\) \(\approx\) \(0.247023 - 0.492408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 \)
good7 \( 1 + (-0.896 + 3.34i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.896 - 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.12 + 2.12i)T - 17iT^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 + (5.79 - 1.55i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.73 + 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 + (10.5 + 6.06i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.01 + 1.34i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.79 - 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.67 + 0.448i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (-6.12 + 6.12i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.10 + 11.5i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + (2.24 - 8.36i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.70297598696944194184856813825, −10.13721942417965969298715100309, −9.135179455923880525518909836740, −7.73262490835372580561117062276, −7.24346126452067569972904129949, −6.31609272024795873757017328064, −4.97245233580033949985592584252, −3.96715444969358188815098036970, −1.99133345595986607385407332683, −0.49338548179738134441149647277, 1.62370666241228614173726748757, 3.38991426959229186858895279610, 5.15467966955851108580759574205, 5.73590578150781590081588451479, 6.58715281829311741703900575518, 8.090973362578471386152467801454, 8.460947437852910368841159414995, 9.957077405498022777273325351546, 10.28025899671369381570331309790, 11.32344547180921816322873970169

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