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Properties

Label 2-450-9.4-c1-0-10
Degree $2$
Conductor $450$
Sign $0.861 + 0.507i$
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 + (1.72 + 0.158i)3-s + (−0.499 + 0.866i)4-s + (−0.724 − 1.57i)6-s + (−0.224 − 0.389i)7-s + 0.999·8-s + (2.94 + 0.548i)9-s + (1.72 + 2.98i)11-s + (−0.999 + 1.41i)12-s + (1.22 − 2.12i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s + 5.89·17-s + (−0.999 − 2.82i)18-s − 5.44·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.995 + 0.0917i)3-s + (−0.249 + 0.433i)4-s + (−0.295 − 0.642i)6-s + (−0.0849 − 0.147i)7-s + 0.353·8-s + (0.983 + 0.182i)9-s + (0.520 + 0.900i)11-s + (−0.288 + 0.408i)12-s + (0.339 − 0.588i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s + 1.43·17-s + (−0.235 − 0.666i)18-s − 1.25·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.861 + 0.507i$
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: \((2,\ 450,\ (\ :1/2),\ 0.861 + 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62265 - 0.442823i\)
\(L(\frac12)\) \(\approx\) \(1.62265 - 0.442823i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.158i)T \)
5 \( 1 \)
good7 \( 1 + (0.224 + 0.389i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.22 + 2.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
23 \( 1 + (-3.44 + 5.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.22 + 3.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + (6.62 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.27 + 3.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.44T + 71T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + (3.67 + 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47019347343418021504273947881, −10.35984364419677795149384479556, −9.137028979010818651386532209918, −8.539165879378360811089680810398, −7.57460002899200076740563201796, −6.64861034249493424035442505697, −4.90778985382593298254856423134, −3.83203018691601258544443203384, −2.83405563618149958920171559143, −1.48144503924421526760917687131, 1.46610011522956910047700798362, 3.16001830945479658496128919553, 4.23195451201928269220460429662, 5.71574852324829156049971928202, 6.66120356212894529150396452585, 7.65904967431536963195767510084, 8.471929667688983269707444755947, 9.150148148039742460572913714634, 9.939917889334503721337988729519, 11.00212099086590421932900560275

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