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Properties

Label 2-405-5.3-c2-0-24
Degree $2$
Conductor $405$
Sign $0.848 + 0.529i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.54i)2-s − 0.766i·4-s + (−4.99 − 0.0220i)5-s + (1.44 − 1.44i)7-s + (4.99 + 4.99i)8-s + (−7.75 + 7.68i)10-s + 12.6·11-s + (9.41 + 9.41i)13-s − 4.45i·14-s + 18.4·16-s + (16.3 − 16.3i)17-s + 9.12i·19-s + (−0.0168 + 3.83i)20-s + (19.5 − 19.5i)22-s + (−15.6 − 15.6i)23-s + ⋯
L(s)  = 1  + (0.771 − 0.771i)2-s − 0.191i·4-s + (−0.999 − 0.00440i)5-s + (0.206 − 0.206i)7-s + (0.624 + 0.624i)8-s + (−0.775 + 0.768i)10-s + 1.15·11-s + (0.724 + 0.724i)13-s − 0.318i·14-s + 1.15·16-s + (0.960 − 0.960i)17-s + 0.480i·19-s + (−0.000843 + 0.191i)20-s + (0.888 − 0.888i)22-s + (−0.680 − 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.43895 - 0.698659i\)
\(L(\frac12)\) \(\approx\) \(2.43895 - 0.698659i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (4.99 + 0.0220i)T \)
good2 \( 1 + (-1.54 + 1.54i)T - 4iT^{2} \)
7 \( 1 + (-1.44 + 1.44i)T - 49iT^{2} \)
11 \( 1 - 12.6T + 121T^{2} \)
13 \( 1 + (-9.41 - 9.41i)T + 169iT^{2} \)
17 \( 1 + (-16.3 + 16.3i)T - 289iT^{2} \)
19 \( 1 - 9.12iT - 361T^{2} \)
23 \( 1 + (15.6 + 15.6i)T + 529iT^{2} \)
29 \( 1 + 19.2iT - 841T^{2} \)
31 \( 1 - 18.6T + 961T^{2} \)
37 \( 1 + (32.6 - 32.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 51.6T + 1.68e3T^{2} \)
43 \( 1 + (-33.7 - 33.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-37.6 + 37.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (18.2 + 18.2i)T + 2.80e3iT^{2} \)
59 \( 1 - 99.2iT - 3.48e3T^{2} \)
61 \( 1 + 103.T + 3.72e3T^{2} \)
67 \( 1 + (-13.8 + 13.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 54.5T + 5.04e3T^{2} \)
73 \( 1 + (8.13 + 8.13i)T + 5.32e3iT^{2} \)
79 \( 1 + 141. iT - 6.24e3T^{2} \)
83 \( 1 + (48.1 + 48.1i)T + 6.88e3iT^{2} \)
89 \( 1 - 59.3iT - 7.92e3T^{2} \)
97 \( 1 + (80.4 - 80.4i)T - 9.40e3iT^{2} \)
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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

−11.30258664291272376397056368957, −10.42359325387661105968127379063, −9.143991307715348698026346601739, −8.130246202668273037938016169915, −7.31584235530669841941832687341, −6.05566648807187177416611287823, −4.52418798233843131869320031972, −4.02224179544153418817079117450, −2.96733344425072320000905195623, −1.28426959938822004009626765995, 1.17117014731957850624347405170, 3.51696902882920631169565023675, 4.18144458028204346572557604988, 5.43824716623833327993032266679, 6.24757341731921298131063287381, 7.29567738067467517885168372430, 8.075225088501008561348567433323, 9.102621706443459904491527297669, 10.38300804916284267082593860459, 11.18838174726473187132362111677

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