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Properties

Label 2-450-9.4-c1-0-3
Degree $2$
Conductor $450$
Sign $-0.939 - 0.342i$
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 + 1.73i·3-s + (−0.499 + 0.866i)4-s + (−1.49 + 0.866i)6-s + (1 + 1.73i)7-s − 0.999·8-s − 2.99·9-s + (−1.49 − 0.866i)12-s + (−2 + 3.46i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s + 6·17-s + (−1.49 − 2.59i)18-s − 7·19-s + (−2.99 + 1.73i)21-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 0.999i·3-s + (−0.249 + 0.433i)4-s + (−0.612 + 0.353i)6-s + (0.377 + 0.654i)7-s − 0.353·8-s − 0.999·9-s + (−0.433 − 0.249i)12-s + (−0.554 + 0.960i)13-s + (−0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s + 1.45·17-s + (−0.353 − 0.612i)18-s − 1.60·19-s + (−0.654 + 0.377i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.939 - 0.342i$
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: \((2,\ 450,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253086 + 1.43532i\)
\(L(\frac12)\) \(\approx\) \(0.253086 + 1.43532i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.65472466997933109799150388013, −10.47174748901983443790504414975, −9.625778802538188999736617137024, −8.712950734487194954332160072320, −8.020429779086631399442985567981, −6.66298818596799400334007289537, −5.67769869635210712620004877875, −4.80672030919431728779799726607, −3.91434509398070000055591273648, −2.51849519034468010214177899111, 0.834927215286127367792479834614, 2.27912390423606772154902281763, 3.49986925564631839877185963284, 4.89793932327320862923762674513, 5.90281402754987293962748121180, 7.00482223282437383387882383757, 7.939291179382744755355383843969, 8.716621750390633761285193056045, 10.27846725553952859944396915196, 10.57021510800665011692791231956

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