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Properties

Label 2-450-225.79-c1-0-2
Degree $2$
Conductor $450$
Sign $-0.418 - 0.908i$
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 + (1.55 + 0.759i)3-s + (0.104 − 0.994i)4-s + (−2.15 + 0.585i)5-s + (−1.66 + 0.477i)6-s + (0.358 + 0.206i)7-s + (0.587 + 0.809i)8-s + (1.84 + 2.36i)9-s + (1.21 − 1.87i)10-s + (2.71 + 3.00i)11-s + (0.917 − 1.46i)12-s + (−2.60 − 2.34i)13-s + (−0.404 + 0.0859i)14-s + (−3.80 − 0.726i)15-s + (−0.978 − 0.207i)16-s + (4.00 + 5.51i)17-s + ⋯
L(s)  = 1  + (−0.525 + 0.473i)2-s + (0.898 + 0.438i)3-s + (0.0522 − 0.497i)4-s + (−0.965 + 0.261i)5-s + (−0.679 + 0.194i)6-s + (0.135 + 0.0781i)7-s + (0.207 + 0.286i)8-s + (0.615 + 0.787i)9-s + (0.383 − 0.594i)10-s + (0.817 + 0.907i)11-s + (0.264 − 0.424i)12-s + (−0.721 − 0.649i)13-s + (−0.108 + 0.0229i)14-s + (−0.982 − 0.187i)15-s + (−0.244 − 0.0519i)16-s + (0.971 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.418 - 0.908i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.633382 + 0.989751i\)
\(L(\frac12)\) \(\approx\) \(0.633382 + 0.989751i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 - 0.759i)T \)
5 \( 1 + (2.15 - 0.585i)T \)
good7 \( 1 + (-0.358 - 0.206i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.71 - 3.00i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (2.60 + 2.34i)T + (1.35 + 12.9i)T^{2} \)
17 \( 1 + (-4.00 - 5.51i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.01 - 3.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.423 - 1.99i)T + (-21.0 + 9.35i)T^{2} \)
29 \( 1 + (7.84 - 3.49i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (3.32 + 1.47i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-10.4 + 3.39i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.65 + 2.95i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (-6.62 - 3.82i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.05 + 6.85i)T + (-31.4 + 34.9i)T^{2} \)
53 \( 1 + (-1.02 + 1.40i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.30 - 7.00i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-8.19 - 9.09i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (-2.28 + 5.13i)T + (-44.8 - 49.7i)T^{2} \)
71 \( 1 + (-0.156 - 0.113i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.97 + 1.94i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.31 + 3.70i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-10.1 + 1.07i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (-1.79 + 5.51i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.26 + 5.08i)T + (-64.9 + 72.0i)T^{2} \)
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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

−11.07321165243353008640497214612, −10.27833723628981645607495698826, −9.507121075901617924784064827215, −8.541477451469339887442698786632, −7.74777809475593722895658948543, −7.24277936125012108686393050560, −5.80115506097554640373216273685, −4.38261996748796398899820097038, −3.58686447432921755001451917373, −1.92856275557698587318038118079, 0.824851779515445120002769204424, 2.50622953112754725525811736342, 3.62459146626572999484005693303, 4.58677518678132451832816302886, 6.50610795307578124172489247671, 7.49274775633787658888095945371, 8.064620912237185198154107342978, 9.187979406049764291455832794043, 9.423831286977915772636080989568, 11.04852887984761897719696784288

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