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Properties

Label 2-450-5.4-c3-0-14
Degree $2$
Conductor $450$
Sign $0.894 + 0.447i$
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 + 14i·7-s − 8i·8-s − 6·11-s − 68i·13-s − 28·14-s + 16·16-s − 78i·17-s − 44·19-s − 12i·22-s + 120i·23-s + 136·26-s − 56i·28-s + 126·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.755i·7-s − 0.353i·8-s − 0.164·11-s − 1.45i·13-s − 0.534·14-s + 0.250·16-s − 1.11i·17-s − 0.531·19-s − 0.116i·22-s + 1.08i·23-s + 1.02·26-s − 0.377i·28-s + 0.806·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 450,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.303122503\)
\(L(\frac12)\) \(\approx\) \(1.303122503\)
\(L(\frac{5}{2})\) 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 \( 1 - 14iT - 343T^{2} \)
11 \( 1 + 6T + 1.33e3T^{2} \)
13 \( 1 + 68iT - 2.19e3T^{2} \)
17 \( 1 + 78iT - 4.91e3T^{2} \)
19 \( 1 + 44T + 6.85e3T^{2} \)
23 \( 1 - 120iT - 1.21e4T^{2} \)
29 \( 1 - 126T + 2.43e4T^{2} \)
31 \( 1 + 244T + 2.97e4T^{2} \)
37 \( 1 + 304iT - 5.06e4T^{2} \)
41 \( 1 - 480T + 6.89e4T^{2} \)
43 \( 1 + 104iT - 7.95e4T^{2} \)
47 \( 1 + 600iT - 1.03e5T^{2} \)
53 \( 1 + 258iT - 1.48e5T^{2} \)
59 \( 1 - 534T + 2.05e5T^{2} \)
61 \( 1 - 362T + 2.26e5T^{2} \)
67 \( 1 + 268iT - 3.00e5T^{2} \)
71 \( 1 - 972T + 3.57e5T^{2} \)
73 \( 1 + 470iT - 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 396iT - 5.71e5T^{2} \)
89 \( 1 + 972T + 7.04e5T^{2} \)
97 \( 1 + 46iT - 9.12e5T^{2} \)
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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

−10.50660331137782613243107151089, −9.515525246392198693287779669934, −8.727113518480559458849071489633, −7.79970490284372622912019963914, −6.99354451089136047503674363806, −5.64816651530711963779472048450, −5.28035131384639514186325059989, −3.74001390302376578162910707036, −2.44919589353584210765304264684, −0.46752539479720959482519751673, 1.19952123479345551092621614185, 2.45923174085190700919372007706, 3.95883014500040833368426778375, 4.53450362904211865588275318406, 6.06993041983229543660622433247, 7.02497890527338251595744714774, 8.199210416913976268165464284598, 9.042088177048726839310535444920, 10.02169034745124019765302137638, 10.79055908212195378412944646315

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