Properties

Label 2-405-5.4-c3-0-43
Degree $2$
Conductor $405$
Sign $-0.00692 + 0.999i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
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  − 2.46i·2-s + 1.92·4-s + (−0.0773 + 11.1i)5-s − 19.2i·7-s − 24.4i·8-s + (27.5 + 0.190i)10-s + 39.8·11-s − 1.00i·13-s − 47.4·14-s − 44.8·16-s + 52.6i·17-s + 49.5·19-s + (−0.149 + 21.5i)20-s − 98.1i·22-s − 27.4i·23-s + ⋯
L(s)  = 1  − 0.871i·2-s + 0.241·4-s + (−0.00692 + 0.999i)5-s − 1.03i·7-s − 1.08i·8-s + (0.871 + 0.00602i)10-s + 1.09·11-s − 0.0215i·13-s − 0.904·14-s − 0.700·16-s + 0.750i·17-s + 0.598·19-s + (−0.00166 + 0.241i)20-s − 0.951i·22-s − 0.248i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00692 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00692 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.00692 + 0.999i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.00692 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.336482228\)
\(L(\frac12)\) \(\approx\) \(2.336482228\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.0773 - 11.1i)T \)
good2 \( 1 + 2.46iT - 8T^{2} \)
7 \( 1 + 19.2iT - 343T^{2} \)
11 \( 1 - 39.8T + 1.33e3T^{2} \)
13 \( 1 + 1.00iT - 2.19e3T^{2} \)
17 \( 1 - 52.6iT - 4.91e3T^{2} \)
19 \( 1 - 49.5T + 6.85e3T^{2} \)
23 \( 1 + 27.4iT - 1.21e4T^{2} \)
29 \( 1 - 254.T + 2.43e4T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + 419. iT - 5.06e4T^{2} \)
41 \( 1 - 398.T + 6.89e4T^{2} \)
43 \( 1 + 358. iT - 7.95e4T^{2} \)
47 \( 1 - 141. iT - 1.03e5T^{2} \)
53 \( 1 + 290. iT - 1.48e5T^{2} \)
59 \( 1 + 28.7T + 2.05e5T^{2} \)
61 \( 1 - 732.T + 2.26e5T^{2} \)
67 \( 1 + 176. iT - 3.00e5T^{2} \)
71 \( 1 + 802.T + 3.57e5T^{2} \)
73 \( 1 + 512. iT - 3.89e5T^{2} \)
79 \( 1 + 612.T + 4.93e5T^{2} \)
83 \( 1 + 80.8iT - 5.71e5T^{2} \)
89 \( 1 + 24.0T + 7.04e5T^{2} \)
97 \( 1 - 1.36e3iT - 9.12e5T^{2} \)
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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

−10.64182650991123660137821934828, −10.12604780067578397862215562934, −9.079111472767069083446030015500, −7.54284651569219664048277231855, −6.91755358740000656758807866418, −6.04660230080694100052693840947, −4.10320091835147792335277754195, −3.49661971064806961282178277765, −2.19416111028581988548156162931, −0.876265744764014218158232557470, 1.32334815744951814316955701697, 2.78238063478243460942804420637, 4.53268883990074064686608654483, 5.47143291994441465669819914415, 6.25035510297173262760152230225, 7.28764020663213363376763941744, 8.358329935811511360189085992193, 8.959599560922391996534800441606, 9.826133357862015082617072859184, 11.48928059739746581391628738078

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