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Properties

Label 2-450-9.4-c1-0-4
Degree $2$
Conductor $450$
Sign $0.635 - 0.771i$
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 + 1.41i)3-s + (−0.499 + 0.866i)4-s + (0.724 − 1.57i)6-s + (−0.224 − 0.389i)7-s + 0.999·8-s + (−1.00 + 2.82i)9-s + (2.44 + 4.24i)11-s + (−1.72 + 0.158i)12-s + (−0.224 + 0.389i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s − 4.89·17-s + (2.94 − 0.548i)18-s + 7.44·19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.577 + 0.816i)3-s + (−0.249 + 0.433i)4-s + (0.295 − 0.642i)6-s + (−0.0849 − 0.147i)7-s + 0.353·8-s + (−0.333 + 0.942i)9-s + (0.738 + 1.27i)11-s + (−0.497 + 0.0458i)12-s + (−0.0623 + 0.107i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s − 1.18·17-s + (0.695 − 0.129i)18-s + 1.70·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.635 - 0.771i$
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: \((2,\ 450,\ (\ :1/2),\ 0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19691 + 0.564869i\)
\(L(\frac12)\) \(\approx\) \(1.19691 + 0.564869i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (0.224 + 0.389i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.224 - 0.389i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.27 - 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.44 + 9.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.55T + 53T^{2} \)
59 \( 1 + (-2.72 + 4.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.174 + 0.301i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (8.34 + 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.72 + 4.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-4.39 - 7.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.20533555646181139647924274720, −10.00185841625728219609381236089, −9.629515532663693523951460440062, −8.826096697357753302317035887218, −7.76033775116971610525789240604, −6.81326868612938860292858465141, −5.08514616468498687969914573157, −4.22518941972148503764035812523, −3.17160584295484426980374790533, −1.84409638398667191043085923391, 0.940478734673988898420244432043, 2.65906363954321131741003417277, 4.00014893588602093027853337293, 5.67364164034654748784704880143, 6.40459614261678717758318989290, 7.34984464884695018596445258736, 8.211436171680862819264980752986, 8.998262613818015432743854519110, 9.638167041789377787053680840713, 11.09802780315976911152573365255

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