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Properties

Label 2-450-9.7-c1-0-10
Degree $2$
Conductor $450$
Sign $-0.173 + 0.984i$
Analytic cond. $3.59326$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 1.73i·6-s + (−2 + 3.46i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−1.5 + 2.59i)11-s + (1.49 + 0.866i)12-s + (−2 − 3.46i)13-s + (−1.99 − 3.46i)14-s + (−0.5 + 0.866i)16-s + 3·17-s + (1.5 + 2.59i)18-s − 4·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.707i·6-s + (−0.755 + 1.30i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.452 + 0.783i)11-s + (0.433 + 0.250i)12-s + (−0.554 − 0.960i)13-s + (−0.534 − 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.727·17-s + (0.353 + 0.612i)18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 \)
good7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.41929159624427283601670106434, −10.05049605378469152312790453051, −9.144004254066264603035661950833, −8.139305372682385888948115746165, −6.98160606473388361582558565131, −5.94731165742957080282610511362, −5.45367246275482720453311730346, −4.26574334934088432481035944222, −2.54315119257022247051796027279, 0, 1.53972805685923414681871814262, 3.30859819364722614495584263813, 4.45620166140926607087639742337, 5.73465248675527525835266042215, 6.93152735287150688178153848895, 7.45385497477481971104160936022, 8.703415464149144452575373545354, 9.909530277399859716106180001218, 10.54911184061948457325943695062

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