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Properties

Label 2-450-25.9-c1-0-10
Degree $2$
Conductor $450$
Sign $-0.959 + 0.280i$
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 + (−2.23 + 0.107i)5-s − 0.992i·7-s + (−0.951 − 0.309i)8-s + (−1.22 + 1.87i)10-s + (−2.58 − 1.87i)11-s + (−2.26 − 3.11i)13-s + (−0.802 − 0.583i)14-s + (−0.809 + 0.587i)16-s + (−2.15 − 0.701i)17-s + (1.45 − 4.46i)19-s + (0.792 + 2.09i)20-s + (−3.03 + 0.986i)22-s + (−3.29 + 4.53i)23-s + ⋯
L(s)  = 1  + (0.415 − 0.572i)2-s + (−0.154 − 0.475i)4-s + (−0.998 + 0.0481i)5-s − 0.375i·7-s + (−0.336 − 0.109i)8-s + (−0.387 + 0.591i)10-s + (−0.778 − 0.565i)11-s + (−0.628 − 0.864i)13-s + (−0.214 − 0.155i)14-s + (−0.202 + 0.146i)16-s + (−0.523 − 0.170i)17-s + (0.332 − 1.02i)19-s + (0.177 + 0.467i)20-s + (−0.647 + 0.210i)22-s + (−0.686 + 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.959 + 0.280i$
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: \((2,\ 450,\ (\ :1/2),\ -0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118918 - 0.829351i\)
\(L(\frac12)\) \(\approx\) \(0.118918 - 0.829351i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.107i)T \)
good7 \( 1 + 0.992iT - 7T^{2} \)
11 \( 1 + (2.58 + 1.87i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.26 + 3.11i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.15 + 0.701i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.45 + 4.46i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.29 - 4.53i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.25 + 6.95i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.07 - 4.22i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.919 + 0.667i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.88iT - 43T^{2} \)
47 \( 1 + (-6.88 + 2.23i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.00 - 1.30i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.55 + 4.03i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.95 - 6.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (12.4 + 4.04i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.675 + 2.07i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.67 + 2.30i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.44 + 10.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-9.87 - 3.20i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.56 + 1.13i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (15.7 - 5.11i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.82811729797485151092274582783, −10.07943206369845465153326898532, −8.905854229255572141661204100625, −7.85790202529732740540601932454, −7.12972889293753969001925649109, −5.67274395277339263127399118550, −4.71884707173453753378389780647, −3.63171411173450552977727641368, −2.61673828185607706764697162318, −0.43963313846975486563095110210, 2.41703664277798089766013627330, 3.91136378608272610535770732181, 4.70285495603406528096714521342, 5.82063893202423724543252016790, 7.01898255380829858653815369061, 7.70333266241310975534281492933, 8.561428697099953059469119385524, 9.563510595991762989227758650258, 10.73118491495218078131991888407, 11.70491156781973667119356166791

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