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Properties

Label 2-450-5.4-c3-0-16
Degree $2$
Conductor $450$
Sign $0.447 + 0.894i$
Analytic cond. $26.5508$
Root an. cond. $5.15275$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 4·4-s − i·7-s − 8i·8-s − 42·11-s + 67i·13-s + 2·14-s + 16·16-s − 54i·17-s + 115·19-s − 84i·22-s − 162i·23-s − 134·26-s + 4i·28-s − 210·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.0539i·7-s − 0.353i·8-s − 1.15·11-s + 1.42i·13-s + 0.0381·14-s + 0.250·16-s − 0.770i·17-s + 1.38·19-s − 0.814i·22-s − 1.46i·23-s − 1.01·26-s + 0.0269i·28-s − 1.34·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(26.5508\)
Root analytic conductor: \(5.15275\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8325663909\)
\(L(\frac12)\) \(\approx\) \(0.8325663909\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT - 343T^{2} \)
11 \( 1 + 42T + 1.33e3T^{2} \)
13 \( 1 - 67iT - 2.19e3T^{2} \)
17 \( 1 + 54iT - 4.91e3T^{2} \)
19 \( 1 - 115T + 6.85e3T^{2} \)
23 \( 1 + 162iT - 1.21e4T^{2} \)
29 \( 1 + 210T + 2.43e4T^{2} \)
31 \( 1 + 193T + 2.97e4T^{2} \)
37 \( 1 + 286iT - 5.06e4T^{2} \)
41 \( 1 + 12T + 6.89e4T^{2} \)
43 \( 1 + 263iT - 7.95e4T^{2} \)
47 \( 1 + 414iT - 1.03e5T^{2} \)
53 \( 1 + 192iT - 1.48e5T^{2} \)
59 \( 1 - 690T + 2.05e5T^{2} \)
61 \( 1 + 733T + 2.26e5T^{2} \)
67 \( 1 - 299iT - 3.00e5T^{2} \)
71 \( 1 - 228T + 3.57e5T^{2} \)
73 \( 1 + 938iT - 3.89e5T^{2} \)
79 \( 1 - 160T + 4.93e5T^{2} \)
83 \( 1 + 462iT - 5.71e5T^{2} \)
89 \( 1 + 240T + 7.04e5T^{2} \)
97 \( 1 + 511iT - 9.12e5T^{2} \)
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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

−10.42358364976021129169946794179, −9.402775764559550739188966706582, −8.739591927397558823403244219767, −7.48998783759318748165935589343, −7.03886836608338327275433017618, −5.71360486807984569434128470588, −4.93138856214640589143034796965, −3.73581182558786197460205492013, −2.20996701283792806053974829948, −0.28117832935911331330552939217, 1.28978331279074381337272596413, 2.78991282878349325069165891270, 3.65225842048490600129525126996, 5.20584491168773341484575028387, 5.72425587523346107784076344787, 7.52123912447999566358368013415, 8.004239565543575578434790469912, 9.276509670919285682082372819313, 10.04631242273619391487746896641, 10.83873681475319718413332433348

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