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Properties

Label 2-450-9.4-c1-0-12
Degree $2$
Conductor $450$
Sign $-0.00922 + 0.999i$
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  + (−0.5 − 0.866i)2-s + (1 − 1.41i)3-s + (−0.499 + 0.866i)4-s + (−1.72 − 0.158i)6-s + (2.22 + 3.85i)7-s + 0.999·8-s + (−1.00 − 2.82i)9-s + (−2.44 − 4.24i)11-s + (0.724 + 1.57i)12-s + (2.22 − 3.85i)13-s + (2.22 − 3.85i)14-s + (−0.5 − 0.866i)16-s + 4.89·17-s + (−1.94 + 2.28i)18-s + 2.55·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.577 − 0.816i)3-s + (−0.249 + 0.433i)4-s + (−0.704 − 0.0648i)6-s + (0.840 + 1.45i)7-s + 0.353·8-s + (−0.333 − 0.942i)9-s + (−0.738 − 1.27i)11-s + (0.209 + 0.454i)12-s + (0.617 − 1.06i)13-s + (0.594 − 1.02i)14-s + (−0.125 − 0.216i)16-s + 1.18·17-s + (−0.459 + 0.537i)18-s + 0.585·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00922 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.00922 + 0.999i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.00922 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04005 - 1.04970i\)
\(L(\frac12)\) \(\approx\) \(1.04005 - 1.04970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (-2.22 - 3.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.22 + 3.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.224 + 0.389i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.72 - 6.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.550 + 0.953i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.44T + 53T^{2} \)
59 \( 1 + (-0.275 + 0.476i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.17 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.34T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.275 + 0.476i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (5.39 + 9.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.08092400030641641054492514932, −9.911856113364107318260143285216, −8.764034209161510767661851162484, −8.280541912827560311698979564867, −7.69820320809553272417945105231, −6.00134730933954020510231946879, −5.32505731276368918576676120858, −3.30060305807911241307149981494, −2.62062605954769171436461843715, −1.14452579240521384032137373422, 1.68905030835612399919266939780, 3.66986786430764568360603877297, 4.57617253747426899220778771902, 5.40086759351893631576848613960, 7.21946268323579942959578671100, 7.50919475845530423245898231414, 8.570604184693906623261933787922, 9.542961225881127462479407720895, 10.32741901593949830487861308336, 10.84361118827544965254166606666

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