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Properties

Label 2-450-5.4-c5-0-25
Degree $2$
Conductor $450$
Sign $0.894 + 0.447i$
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 − 172i·7-s − 64i·8-s − 132·11-s + 946i·13-s + 688·14-s + 256·16-s + 222i·17-s − 500·19-s − 528i·22-s + 3.56e3i·23-s − 3.78e3·26-s + 2.75e3i·28-s + 2.19e3·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.32i·7-s − 0.353i·8-s − 0.328·11-s + 1.55i·13-s + 0.938·14-s + 0.250·16-s + 0.186i·17-s − 0.317·19-s − 0.232i·22-s + 1.40i·23-s − 1.09·26-s + 0.663i·28-s + 0.483·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(72.1727\)
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: \((2,\ 450,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.448354489\)
\(L(\frac12)\) \(\approx\) \(1.448354489\)
\(L(\frac{7}{2})\) 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 \( 1 + 172iT - 1.68e4T^{2} \)
11 \( 1 + 132T + 1.61e5T^{2} \)
13 \( 1 - 946iT - 3.71e5T^{2} \)
17 \( 1 - 222iT - 1.41e6T^{2} \)
19 \( 1 + 500T + 2.47e6T^{2} \)
23 \( 1 - 3.56e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.19e3T + 2.05e7T^{2} \)
31 \( 1 - 2.31e3T + 2.86e7T^{2} \)
37 \( 1 + 1.12e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.24e3T + 1.15e8T^{2} \)
43 \( 1 + 2.06e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.58e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.10e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.98e3T + 7.14e8T^{2} \)
61 \( 1 - 1.66e4T + 8.44e8T^{2} \)
67 \( 1 - 1.80e3iT - 1.35e9T^{2} \)
71 \( 1 - 2.45e4T + 1.80e9T^{2} \)
73 \( 1 + 2.04e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.62e4T + 3.07e9T^{2} \)
83 \( 1 + 5.15e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.10e5T + 5.58e9T^{2} \)
97 \( 1 + 7.83e4iT - 8.58e9T^{2} \)
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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

−10.11697058626191480572388079006, −9.270619998556645550580208910379, −8.273541276897703962152377969164, −7.23705098257788658970033089901, −6.78512394185472005897638334966, −5.53723246327256231871992359949, −4.38369306649188489267908197685, −3.67157197122802581019873530613, −1.81754458528218558292932608072, −0.43007168670062072096296076288, 0.911619783185712865114913543018, 2.46942471888206293635502685017, 3.01968227939375626281256026143, 4.60139646379683965057252500997, 5.48388134618057891145764387822, 6.42387844313989723250318154235, 8.046282784532476610438762097517, 8.504965912789616761636397457195, 9.600768136078730718718491891313, 10.38858643790891307381281046454

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