Properties

Label 2-450-5.4-c3-0-8
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 − 11i·7-s − 8i·8-s − 36·11-s + 17i·13-s + 22·14-s + 16·16-s + 12i·17-s + 91·19-s − 72i·22-s + 60i·23-s − 34·26-s + 44i·28-s + 276·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.593i·7-s − 0.353i·8-s − 0.986·11-s + 0.362i·13-s + 0.419·14-s + 0.250·16-s + 0.171i·17-s + 1.09·19-s − 0.697i·22-s + 0.543i·23-s − 0.256·26-s + 0.296i·28-s + 1.76·29-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

\(L(2)\) \(\approx\) \(1.647289323\)
\(L(\frac12)\) \(\approx\) \(1.647289323\)
\(L(\frac{5}{2})\) 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 \( 1 + 11iT - 343T^{2} \)
11 \( 1 + 36T + 1.33e3T^{2} \)
13 \( 1 - 17iT - 2.19e3T^{2} \)
17 \( 1 - 12iT - 4.91e3T^{2} \)
19 \( 1 - 91T + 6.85e3T^{2} \)
23 \( 1 - 60iT - 1.21e4T^{2} \)
29 \( 1 - 276T + 2.43e4T^{2} \)
31 \( 1 - 191T + 2.97e4T^{2} \)
37 \( 1 + 254iT - 5.06e4T^{2} \)
41 \( 1 + 60T + 6.89e4T^{2} \)
43 \( 1 + 49iT - 7.95e4T^{2} \)
47 \( 1 - 600iT - 1.03e5T^{2} \)
53 \( 1 - 612iT - 1.48e5T^{2} \)
59 \( 1 - 744T + 2.05e5T^{2} \)
61 \( 1 - 167T + 2.26e5T^{2} \)
67 \( 1 - 457iT - 3.00e5T^{2} \)
71 \( 1 + 588T + 3.57e5T^{2} \)
73 \( 1 + 970iT - 3.89e5T^{2} \)
79 \( 1 + 164T + 4.93e5T^{2} \)
83 \( 1 - 696iT - 5.71e5T^{2} \)
89 \( 1 - 1.24e3T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−10.62284928853726417795088437891, −9.934978587026581263393392126340, −8.911276888816445216283732774295, −7.88638865130561664464652003333, −7.26469945981216460894488234391, −6.18710203317889426590344945192, −5.16380232539005393820449772736, −4.19482502408834331661313655628, −2.83164762257168451306261265075, −0.929758947206530202672196434107, 0.73564221042227265882887672263, 2.38164170879179700850593233672, 3.22762415154801947909160522475, 4.74700347141400335362808966950, 5.49338182863879290846114428314, 6.78293055421717022132271180181, 8.091122402164440803502143178906, 8.664658720501297887862065302921, 9.987175152020956575362982121629, 10.29841567752806645979354562846

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