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Properties

Label 2-450-225.94-c1-0-4
Degree $2$
Conductor $450$
Sign $-0.456 - 0.889i$
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  + (−0.743 − 0.669i)2-s + (−1.69 + 0.349i)3-s + (0.104 + 0.994i)4-s + (0.950 + 2.02i)5-s + (1.49 + 0.875i)6-s + (−0.590 + 0.340i)7-s + (0.587 − 0.809i)8-s + (2.75 − 1.18i)9-s + (0.648 − 2.14i)10-s + (1.29 − 1.43i)11-s + (−0.524 − 1.65i)12-s + (−3.42 + 3.08i)13-s + (0.666 + 0.141i)14-s + (−2.31 − 3.10i)15-s + (−0.978 + 0.207i)16-s + (−0.520 + 0.715i)17-s + ⋯
L(s)  = 1  + (−0.525 − 0.473i)2-s + (−0.979 + 0.201i)3-s + (0.0522 + 0.497i)4-s + (0.425 + 0.905i)5-s + (0.610 + 0.357i)6-s + (−0.223 + 0.128i)7-s + (0.207 − 0.286i)8-s + (0.918 − 0.395i)9-s + (0.204 − 0.676i)10-s + (0.389 − 0.433i)11-s + (−0.151 − 0.476i)12-s + (−0.950 + 0.855i)13-s + (0.178 + 0.0378i)14-s + (−0.598 − 0.800i)15-s + (−0.244 + 0.0519i)16-s + (−0.126 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.456 - 0.889i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248220 + 0.406554i\)
\(L(\frac12)\) \(\approx\) \(0.248220 + 0.406554i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (1.69 - 0.349i)T \)
5 \( 1 + (-0.950 - 2.02i)T \)
good7 \( 1 + (0.590 - 0.340i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.29 + 1.43i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (3.42 - 3.08i)T + (1.35 - 12.9i)T^{2} \)
17 \( 1 + (0.520 - 0.715i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.48 + 1.80i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.03 - 4.88i)T + (-21.0 - 9.35i)T^{2} \)
29 \( 1 + (-0.725 - 0.323i)T + (19.4 + 21.5i)T^{2} \)
31 \( 1 + (5.66 - 2.52i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.232 - 0.0755i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-4.34 - 4.82i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (7.14 - 4.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.95 + 4.38i)T + (-31.4 - 34.9i)T^{2} \)
53 \( 1 + (-3.20 - 4.40i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.75 + 5.28i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (8.00 - 8.89i)T + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (4.75 + 10.6i)T + (-44.8 + 49.7i)T^{2} \)
71 \( 1 + (-6.41 + 4.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.59 - 1.49i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.73 - 2.11i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (8.60 + 0.904i)T + (81.1 + 17.2i)T^{2} \)
89 \( 1 + (-1.48 - 4.58i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.48 + 10.0i)T + (-64.9 - 72.0i)T^{2} \)
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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

−11.25954821258124722343059655700, −10.58769600557074797938120705130, −9.694754118387586043685269814430, −9.124592630948197736902192864157, −7.53045163233419038631220029247, −6.72789250582596804921761659465, −5.92146983666143669067147494947, −4.57574436011741848490047639178, −3.29810392824779816452070656637, −1.81585181396479415974083931125, 0.38834196277089216179459052122, 1.96922915016107552830474068140, 4.34943758157473918662018248169, 5.25515731105560823935378034668, 6.08689314159665003692073677649, 7.04872136862124611466503596385, 7.968230980847667090855589503830, 9.054013697977485093975272172622, 9.983378074790703287123200220793, 10.50595693794140222830802737316

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