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Properties

Label 2-405-45.29-c2-0-12
Degree $2$
Conductor $405$
Sign $0.0495 - 0.998i$
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  + (−1.78 + 3.09i)2-s + (−4.38 − 7.59i)4-s + (4.95 + 0.682i)5-s + (−5.21 − 3.01i)7-s + 17.0·8-s + (−10.9 + 14.1i)10-s + (−13.1 − 7.60i)11-s + (−9.54 + 5.51i)13-s + (18.6 − 10.7i)14-s + (−12.9 + 22.4i)16-s + 26.8·17-s + 15.8·19-s + (−16.5 − 40.6i)20-s + (47.0 − 27.1i)22-s + (5.76 + 9.98i)23-s + ⋯
L(s)  = 1  + (−0.893 + 1.54i)2-s + (−1.09 − 1.89i)4-s + (0.990 + 0.136i)5-s + (−0.745 − 0.430i)7-s + 2.13·8-s + (−1.09 + 1.41i)10-s + (−1.19 − 0.690i)11-s + (−0.734 + 0.424i)13-s + (1.33 − 0.768i)14-s + (−0.809 + 1.40i)16-s + 1.57·17-s + 0.836·19-s + (−0.827 − 2.03i)20-s + (2.13 − 1.23i)22-s + (0.250 + 0.434i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0495 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0495 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.0495 - 0.998i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.0495 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.730785 + 0.695405i\)
\(L(\frac12)\) \(\approx\) \(0.730785 + 0.695405i\)
\(L(2)\) 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 \( 1 + (-4.95 - 0.682i)T \)
good2 \( 1 + (1.78 - 3.09i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (5.21 + 3.01i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (13.1 + 7.60i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.54 - 5.51i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 26.8T + 289T^{2} \)
19 \( 1 - 15.8T + 361T^{2} \)
23 \( 1 + (-5.76 - 9.98i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-47.1 - 27.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-15.7 - 27.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 38.1iT - 1.36e3T^{2} \)
41 \( 1 + (-32.6 + 18.8i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-3.93 - 2.26i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-1.85 + 3.21i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 56.1T + 2.80e3T^{2} \)
59 \( 1 + (-43.3 + 25.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-24.2 + 41.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (60.0 - 34.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 38.5iT - 5.04e3T^{2} \)
73 \( 1 + 27.2iT - 5.32e3T^{2} \)
79 \( 1 + (-39.4 + 68.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (75.3 - 130. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 3.50iT - 7.92e3T^{2} \)
97 \( 1 + (-23.9 - 13.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.57495441778548828114829827951, −10.05206028066748922845699697942, −9.399141278092270708152197351259, −8.399617644080004493524984378938, −7.40958160457681999009056876239, −6.75567860771863725339360972680, −5.65658645245941415652445273439, −5.16124969873738208764370480105, −3.00641942616971661659482499774, −0.881821962372327851909456854186, 0.890593716594954638644419362318, 2.49740614439230574407527000025, 2.95223039759283060120601102681, 4.75463804807524833235626543211, 5.89669841833026864132425373789, 7.50064063236327004304522828961, 8.371704575389568580159768461965, 9.546644321010820413217917933528, 10.01744485270557656033970659152, 10.33411072286918974062801411758

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