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Properties

Label 2-450-9.4-c1-0-2
Degree $2$
Conductor $450$
Sign $0.972 - 0.234i$
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 + (−0.5 − 1.65i)3-s + (−0.499 + 0.866i)4-s + (−1.18 + 1.26i)6-s + (1.68 + 2.92i)7-s + 0.999·8-s + (−2.5 + 1.65i)9-s + (2.18 + 3.78i)11-s + (1.68 + 0.396i)12-s + (−3.37 + 5.84i)13-s + (1.68 − 2.92i)14-s + (−0.5 − 0.866i)16-s + 1.62·17-s + (2.68 + 1.33i)18-s − 2.37·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.957i)3-s + (−0.249 + 0.433i)4-s + (−0.484 + 0.515i)6-s + (0.637 + 1.10i)7-s + 0.353·8-s + (−0.833 + 0.552i)9-s + (0.659 + 1.14i)11-s + (0.486 + 0.114i)12-s + (−0.935 + 1.61i)13-s + (0.450 − 0.780i)14-s + (−0.125 − 0.216i)16-s + 0.394·17-s + (0.633 + 0.314i)18-s − 0.544·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.972 - 0.234i$
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: \((2,\ 450,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.914895 + 0.108749i\)
\(L(\frac12)\) \(\approx\) \(0.914895 + 0.108749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 1.65i)T \)
5 \( 1 \)
good7 \( 1 + (-1.68 - 2.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.37 - 5.84i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 + (-0.686 + 1.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.686 + 1.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.37 - 4.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.81 + 4.87i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.68 + 6.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (2.18 - 3.78i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 3.11T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.68 - 6.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (4.18 + 7.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.53189598257962358659636149635, −10.27600604514101008633412103430, −9.175214154468421574162326571034, −8.606603722883854348259344140649, −7.39557845651719156679207722853, −6.72138483076677685519083495070, −5.37226378574877034614356017304, −4.32873757139216511598232275913, −2.37434490956744227691583248713, −1.74274670551797321247400523076, 0.69330471787648547657972885818, 3.26973320707507731294267519279, 4.39829875938970653391149644738, 5.36962303761258296357876959444, 6.24843277043642241968095167081, 7.58181438181280895180891645024, 8.238907163307847926973934599389, 9.330261125889743056825445556844, 10.18525028245143830452984685336, 10.83293004021258555307037769527

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