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Properties

Label 2-405-5.3-c2-0-12
Degree $2$
Conductor $405$
Sign $0.725 - 0.688i$
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  + (2.18 − 2.18i)2-s − 5.58i·4-s + (−1.27 + 4.83i)5-s + (−8.79 + 8.79i)7-s + (−3.46 − 3.46i)8-s + (7.79 + 13.3i)10-s − 10.4·11-s + (11.5 + 11.5i)13-s + 38.4i·14-s + 7.16·16-s + (1.17 − 1.17i)17-s + 1.74i·19-s + (26.9 + 7.11i)20-s + (−22.9 + 22.9i)22-s + (13.9 + 13.9i)23-s + ⋯
L(s)  = 1  + (1.09 − 1.09i)2-s − 1.39i·4-s + (−0.255 + 0.966i)5-s + (−1.25 + 1.25i)7-s + (−0.433 − 0.433i)8-s + (0.779 + 1.33i)10-s − 0.953·11-s + (0.890 + 0.890i)13-s + 2.74i·14-s + 0.447·16-s + (0.0694 − 0.0694i)17-s + 0.0919i·19-s + (1.34 + 0.355i)20-s + (−1.04 + 1.04i)22-s + (0.606 + 0.606i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.725 - 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.725 - 0.688i$
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: \((2,\ 405,\ (\ :1),\ 0.725 - 0.688i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.85160 + 0.738838i\)
\(L(\frac12)\) \(\approx\) \(1.85160 + 0.738838i\)
\(L(2)\) 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 \( 1 + (1.27 - 4.83i)T \)
good2 \( 1 + (-2.18 + 2.18i)T - 4iT^{2} \)
7 \( 1 + (8.79 - 8.79i)T - 49iT^{2} \)
11 \( 1 + 10.4T + 121T^{2} \)
13 \( 1 + (-11.5 - 11.5i)T + 169iT^{2} \)
17 \( 1 + (-1.17 + 1.17i)T - 289iT^{2} \)
19 \( 1 - 1.74iT - 361T^{2} \)
23 \( 1 + (-13.9 - 13.9i)T + 529iT^{2} \)
29 \( 1 - 6.49iT - 841T^{2} \)
31 \( 1 + 37T + 961T^{2} \)
37 \( 1 + (45.6 - 45.6i)T - 1.36e3iT^{2} \)
41 \( 1 - 39.3T + 1.68e3T^{2} \)
43 \( 1 + (25.7 + 25.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-18.0 + 18.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-64.6 - 64.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 52.3iT - 3.48e3T^{2} \)
61 \( 1 + 0.504T + 3.72e3T^{2} \)
67 \( 1 + (-0.669 + 0.669i)T - 4.48e3iT^{2} \)
71 \( 1 - 60.6T + 5.04e3T^{2} \)
73 \( 1 + (-25.6 - 25.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 114. iT - 6.24e3T^{2} \)
83 \( 1 + (90.6 + 90.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 83.4iT - 7.92e3T^{2} \)
97 \( 1 + (-66.2 + 66.2i)T - 9.40e3iT^{2} \)
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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

−11.32292860162831474199144874068, −10.55833561961867331156177274740, −9.683091758778120272322158145278, −8.619958694173635160622428070857, −7.12128073590577509626940814070, −6.08177298550547931758924389038, −5.28630985286281722347184828581, −3.73724662136983965192936131713, −3.06432261069558822918605070455, −2.12549392110464661334897776367, 0.58984482600507611155775085849, 3.34211095099877279838734470880, 4.09198191530682220694380843349, 5.21213831902996322373624832676, 6.01098702050413232806702968916, 7.10773353949460174013843343703, 7.76633756202251108973122342456, 8.820360372439184175176034222910, 10.09582611624446111663020663133, 10.86840606659462430965449133367

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