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Properties

Label 2-450-5.4-c5-0-34
Degree $2$
Conductor $450$
Sign $-0.447 + 0.894i$
Analytic cond. $72.1727$
Root an. cond. $8.49545$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 16·4-s − 241. i·7-s − 64i·8-s + 653.·11-s − 828. i·13-s + 966.·14-s + 256·16-s − 2.16e3i·17-s − 1.25e3·19-s + 2.61e3i·22-s + 3.74e3i·23-s + 3.31e3·26-s + 3.86e3i·28-s − 2.46e3·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.86i·7-s − 0.353i·8-s + 1.62·11-s − 1.35i·13-s + 1.31·14-s + 0.250·16-s − 1.81i·17-s − 0.797·19-s + 1.15i·22-s + 1.47i·23-s + 0.961·26-s + 0.931i·28-s − 0.544·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(72.1727\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.381033097\)
\(L(\frac12)\) \(\approx\) \(1.381033097\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 241. iT - 1.68e4T^{2} \)
11 \( 1 - 653.T + 1.61e5T^{2} \)
13 \( 1 + 828. iT - 3.71e5T^{2} \)
17 \( 1 + 2.16e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.25e3T + 2.47e6T^{2} \)
23 \( 1 - 3.74e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.46e3T + 2.05e7T^{2} \)
31 \( 1 + 1.89e3T + 2.86e7T^{2} \)
37 \( 1 - 1.05e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.96e3T + 1.15e8T^{2} \)
43 \( 1 + 1.10e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.30e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.73e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.52e4T + 7.14e8T^{2} \)
61 \( 1 - 2.68e3T + 8.44e8T^{2} \)
67 \( 1 - 4.81e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.72e4T + 1.80e9T^{2} \)
73 \( 1 - 6.54e3iT - 2.07e9T^{2} \)
79 \( 1 - 6.50e4T + 3.07e9T^{2} \)
83 \( 1 + 6.25e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.79e4T + 5.58e9T^{2} \)
97 \( 1 - 9.59e4iT - 8.58e9T^{2} \)
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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

−9.830741550889302264095238680922, −9.165728199373448099210708647853, −7.82661720223701006259835297730, −7.25940720883381144569966904992, −6.48290332109705766062280523027, −5.23371439919100636162376420660, −4.15333923998169789058613362733, −3.37338813094663318505183797089, −1.23299644825956744436563576699, −0.34823503427311567116341680661, 1.61597752402863575490185789541, 2.24231753754783027460535899930, 3.71382687121261508912707940402, 4.58780957905680133923913664741, 6.03645951779346264929586872493, 6.50619841728221176377765294045, 8.353461835633917743783480678912, 8.924472508550650954210455797056, 9.459705038672380658460545114153, 10.75813173062375346030663923674

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