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Properties

Label 2-450-5.2-c6-0-28
Degree $2$
Conductor $450$
Sign $-0.229 - 0.973i$
Analytic cond. $103.524$
Root an. cond. $10.1746$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 4i)2-s + 32i·4-s + (362. + 362. i)7-s + (−128 + 128i)8-s + 1.32e3·11-s + (−254. + 254. i)13-s + 2.90e3i·14-s − 1.02e3·16-s + (4.39e3 + 4.39e3i)17-s − 819. i·19-s + (5.28e3 + 5.28e3i)22-s + (1.11e4 − 1.11e4i)23-s − 2.03e3·26-s + (−1.16e4 + 1.16e4i)28-s + 2.74e4i·29-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + 0.5i·4-s + (1.05 + 1.05i)7-s + (−0.250 + 0.250i)8-s + 0.992·11-s + (−0.115 + 0.115i)13-s + 1.05i·14-s − 0.250·16-s + (0.895 + 0.895i)17-s − 0.119i·19-s + (0.496 + 0.496i)22-s + (0.915 − 0.915i)23-s − 0.115·26-s + (−0.528 + 0.528i)28-s + 1.12i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(450\)    =    \(2 \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.229 - 0.973i$
Analytic conductor: \(103.524\)
Root analytic conductor: \(10.1746\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{450} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 450,\ (\ :3),\ -0.229 - 0.973i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.901708181\)
\(L(\frac12)\) \(\approx\) \(3.901708181\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-362. - 362. i)T + 1.17e5iT^{2} \)
11 \( 1 - 1.32e3T + 1.77e6T^{2} \)
13 \( 1 + (254. - 254. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-4.39e3 - 4.39e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 819. iT - 4.70e7T^{2} \)
23 \( 1 + (-1.11e4 + 1.11e4i)T - 1.48e8iT^{2} \)
29 \( 1 - 2.74e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.50e4T + 8.87e8T^{2} \)
37 \( 1 + (2.15e4 + 2.15e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 4.56e4T + 4.75e9T^{2} \)
43 \( 1 + (-7.81e4 + 7.81e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (1.99e3 + 1.99e3i)T + 1.07e10iT^{2} \)
53 \( 1 + (-1.06e5 + 1.06e5i)T - 2.21e10iT^{2} \)
59 \( 1 + 6.44e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.20e5T + 5.15e10T^{2} \)
67 \( 1 + (-3.50e4 - 3.50e4i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.51e5T + 1.28e11T^{2} \)
73 \( 1 + (3.26e5 - 3.26e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 2.57e4iT - 2.43e11T^{2} \)
83 \( 1 + (1.23e5 - 1.23e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 1.98e5iT - 4.96e11T^{2} \)
97 \( 1 + (-7.38e5 - 7.38e5i)T + 8.32e11iT^{2} \)
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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

−10.49899875136955644813774903029, −9.063149947254500167899701137658, −8.599920332186315433056078815896, −7.58394142126644643908830394758, −6.53669355065361700236131997863, −5.60041231498892687565561022108, −4.80049265550793102491263147102, −3.70252776374047082903246899760, −2.39743120110000453659291709781, −1.21797566751184326770236933387, 0.804183475403028501775609627457, 1.47940989412192124501967383082, 2.96700992366751523280767277000, 4.08475045999109777530163188242, 4.80804246813951560590907938710, 5.91570995710097906281908362720, 7.16195738301155980816298033170, 7.84948522454693229051838798935, 9.164415648449143602480766607887, 9.974790004982237691257584688752

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