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Properties

Label 2-405-5.4-c3-0-50
Degree $2$
Conductor $405$
Sign $0.940 + 0.340i$
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  + 1.22i·2-s + 6.50·4-s + (10.5 + 3.81i)5-s − 33.1i·7-s + 17.7i·8-s + (−4.65 + 12.8i)10-s + 23.6·11-s − 30.9i·13-s + 40.5·14-s + 30.3·16-s − 48.7i·17-s − 34.3·19-s + (68.3 + 24.8i)20-s + 28.9i·22-s − 181. i·23-s + ⋯
L(s)  = 1  + 0.432i·2-s + 0.813·4-s + (0.940 + 0.340i)5-s − 1.79i·7-s + 0.783i·8-s + (−0.147 + 0.406i)10-s + 0.648·11-s − 0.659i·13-s + 0.773·14-s + 0.474·16-s − 0.695i·17-s − 0.414·19-s + (0.764 + 0.277i)20-s + 0.280i·22-s − 1.64i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.940 + 0.340i$
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: \((2,\ 405,\ (\ :3/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.866888843\)
\(L(\frac12)\) \(\approx\) \(2.866888843\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.5 - 3.81i)T \)
good2 \( 1 - 1.22iT - 8T^{2} \)
7 \( 1 + 33.1iT - 343T^{2} \)
11 \( 1 - 23.6T + 1.33e3T^{2} \)
13 \( 1 + 30.9iT - 2.19e3T^{2} \)
17 \( 1 + 48.7iT - 4.91e3T^{2} \)
19 \( 1 + 34.3T + 6.85e3T^{2} \)
23 \( 1 + 181. iT - 1.21e4T^{2} \)
29 \( 1 - 4.29T + 2.43e4T^{2} \)
31 \( 1 + 270.T + 2.97e4T^{2} \)
37 \( 1 - 163. iT - 5.06e4T^{2} \)
41 \( 1 - 7.40T + 6.89e4T^{2} \)
43 \( 1 + 378. iT - 7.95e4T^{2} \)
47 \( 1 - 177. iT - 1.03e5T^{2} \)
53 \( 1 - 587. iT - 1.48e5T^{2} \)
59 \( 1 - 844.T + 2.05e5T^{2} \)
61 \( 1 - 590.T + 2.26e5T^{2} \)
67 \( 1 - 494. iT - 3.00e5T^{2} \)
71 \( 1 - 53.0T + 3.57e5T^{2} \)
73 \( 1 + 427. iT - 3.89e5T^{2} \)
79 \( 1 - 708.T + 4.93e5T^{2} \)
83 \( 1 + 243. iT - 5.71e5T^{2} \)
89 \( 1 - 558.T + 7.04e5T^{2} \)
97 \( 1 - 471. iT - 9.12e5T^{2} \)
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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

−10.52695431917040353456525653334, −10.26996545107521549264007620546, −8.925807802563330352007965065080, −7.66147712082175035866369777809, −6.92901481112677585705431459699, −6.33216366596228484131048330704, −5.10959289717406789402364298529, −3.72342095774397446648355159827, −2.38095867018075449284932330384, −0.968666573179015198580860718373, 1.66887211142559567352111458851, 2.22255728507847823775159733197, 3.61273747422260010299492829167, 5.35797022151610250420981965963, 6.01310546589645285392206195814, 6.89333707312074026354389926684, 8.380842095028222747389155736150, 9.274857814543801076270634661119, 9.797730223215840737744367769612, 11.12926757631512763233956417879

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