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Properties

Label 2-450-9.4-c1-0-13
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 $0$

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 + (−0.5 − 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−3 − 5.19i)11-s + (1.49 − 0.866i)12-s + (1 − 1.73i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)18-s − 4·19-s + 1.73i·21-s + (3 − 5.19i)22-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 + (−0.188 − 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.904 − 1.56i)11-s + (0.433 − 0.250i)12-s + (0.277 − 0.480i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.353 + 0.612i)18-s − 0.917·19-s + 0.377i·21-s + (0.639 − 1.10i)22-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: $\chi_{450} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615131 - 0.516156i\)
\(L(\frac12)\) \(\approx\) \(0.615131 - 0.516156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84099521265143671295328440686, −10.38884765973097015677744511743, −8.703790023742109870838687063123, −8.043822834862214407046526016415, −6.98324815420559723364351552938, −6.11906377353399424568055431716, −5.43896921001697145723242181190, −4.28338363068696487602464209285, −2.79002133094681193256273376833, −0.49136591243944856185936137948, 1.83284332040228950710394648830, 3.43046685298556412701752143066, 4.65604491669737574417185590289, 5.26922934678786824812404839113, 6.44361951147323624871802717048, 7.42293829254978701231434195071, 9.018186212773898703791761130056, 9.704810257872044344521578935686, 10.57854249578760389817887648176, 11.17925332206604171150886318411

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