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Properties

Label 2-405-5.4-c3-0-13
Degree $2$
Conductor $405$
Sign $-0.695 - 0.718i$
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  + 5.05i·2-s − 17.5·4-s + (−7.77 − 8.03i)5-s − 21.0i·7-s − 48.2i·8-s + (40.6 − 39.2i)10-s − 28.5·11-s − 10.0i·13-s + 106.·14-s + 103.·16-s + 82.7i·17-s − 1.91·19-s + (136. + 141. i)20-s − 144. i·22-s + 170. i·23-s + ⋯
L(s)  = 1  + 1.78i·2-s − 2.19·4-s + (−0.695 − 0.718i)5-s − 1.13i·7-s − 2.13i·8-s + (1.28 − 1.24i)10-s − 0.782·11-s − 0.214i·13-s + 2.02·14-s + 1.61·16-s + 1.18i·17-s − 0.0231·19-s + (1.52 + 1.57i)20-s − 1.39i·22-s + 1.54i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.695 - 0.718i$
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: \((2,\ 405,\ (\ :3/2),\ -0.695 - 0.718i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.054149965\)
\(L(\frac12)\) \(\approx\) \(1.054149965\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.77 + 8.03i)T \)
good2 \( 1 - 5.05iT - 8T^{2} \)
7 \( 1 + 21.0iT - 343T^{2} \)
11 \( 1 + 28.5T + 1.33e3T^{2} \)
13 \( 1 + 10.0iT - 2.19e3T^{2} \)
17 \( 1 - 82.7iT - 4.91e3T^{2} \)
19 \( 1 + 1.91T + 6.85e3T^{2} \)
23 \( 1 - 170. iT - 1.21e4T^{2} \)
29 \( 1 - 256.T + 2.43e4T^{2} \)
31 \( 1 - 48.0T + 2.97e4T^{2} \)
37 \( 1 + 161. iT - 5.06e4T^{2} \)
41 \( 1 - 279.T + 6.89e4T^{2} \)
43 \( 1 - 269. iT - 7.95e4T^{2} \)
47 \( 1 - 9.58iT - 1.03e5T^{2} \)
53 \( 1 - 35.7iT - 1.48e5T^{2} \)
59 \( 1 - 562.T + 2.05e5T^{2} \)
61 \( 1 + 79.3T + 2.26e5T^{2} \)
67 \( 1 + 466. iT - 3.00e5T^{2} \)
71 \( 1 - 316.T + 3.57e5T^{2} \)
73 \( 1 + 633. iT - 3.89e5T^{2} \)
79 \( 1 - 791.T + 4.93e5T^{2} \)
83 \( 1 - 228. iT - 5.71e5T^{2} \)
89 \( 1 + 53.9T + 7.04e5T^{2} \)
97 \( 1 - 96.6iT - 9.12e5T^{2} \)
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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

−11.01951306871507326838916386887, −9.995460925073354026252448191084, −8.933556064154633346178952246472, −7.898564384249970667199602235258, −7.71410016546923808570238076698, −6.57694198397272898656398354836, −5.49671549620286275399319566686, −4.58123683726294772934238094180, −3.72815118604770278075220240337, −0.832809172494836736623136276307, 0.52928745225229150957154582516, 2.51736237018268007910767625693, 2.79276184983220137560925407640, 4.22869943393751783662199609759, 5.18925845673430772489945256336, 6.71083061760969435202528963634, 8.163600557786134555218571311698, 8.867131489252792252697381339761, 9.957510889197530873733812679547, 10.61652488715824305980511600070

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