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Properties

Label 2-450-45.38-c1-0-17
Degree $2$
Conductor $450$
Sign $-0.678 + 0.734i$
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  + (0.965 − 0.258i)2-s + (−0.599 − 1.62i)3-s + (0.866 − 0.499i)4-s + (−1 − 1.41i)6-s + (−1.18 − 4.40i)7-s + (0.707 − 0.707i)8-s + (−2.28 + 1.94i)9-s + (−0.550 − 0.317i)11-s + (−1.33 − 1.10i)12-s + (−0.896 + 3.34i)13-s + (−2.28 − 3.94i)14-s + (0.500 − 0.866i)16-s + (0.317 + 0.317i)17-s + (−1.69 + 2.47i)18-s − 6.44i·19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.346 − 0.938i)3-s + (0.433 − 0.249i)4-s + (−0.408 − 0.577i)6-s + (−0.446 − 1.66i)7-s + (0.249 − 0.249i)8-s + (−0.760 + 0.649i)9-s + (−0.165 − 0.0958i)11-s + (−0.384 − 0.319i)12-s + (−0.248 + 0.928i)13-s + (−0.609 − 1.05i)14-s + (0.125 − 0.216i)16-s + (0.0770 + 0.0770i)17-s + (−0.400 + 0.582i)18-s − 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(3.59326\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626316 - 1.43200i\)
\(L(\frac12)\) \(\approx\) \(0.626316 - 1.43200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.599 + 1.62i)T \)
5 \( 1 \)
good7 \( 1 + (1.18 + 4.40i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.896 - 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.317 - 0.317i)T + 17iT^{2} \)
19 \( 1 + 6.44iT - 19T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.158 + 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-6.39 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.34 - 0.896i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-8.69 + 2.32i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.78 + 3.78i)T - 53iT^{2} \)
59 \( 1 + (-4.48 - 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.38 + 1.71i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.29iT - 71T^{2} \)
73 \( 1 + (-6.89 - 6.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 - 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.41 + 5.26i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 + (0.695 + 2.59i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.00903431187603703752032963846, −10.18813715928946884559516287913, −8.953343641289504889280405875844, −7.45465297329916416917390566472, −7.07512500382379253469826013111, −6.19218499916415024233086095315, −4.90237269610242468376773399513, −3.88661342283345774116408205362, −2.45560667311653101304068440322, −0.822295225431753828815260983995, 2.58262951791814672215436990711, 3.52993927816232735562951592995, 4.88093741403543798096858458831, 5.70249320996759560890557470655, 6.21680573959960329952755707893, 7.83562291427540246681893390356, 8.796553080930884432275303501522, 9.716264879405664038083241383785, 10.52924158415356395340828605724, 11.60373620797483934253148011744

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