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Properties

Label 2-405-45.22-c2-0-26
Degree $2$
Conductor $405$
Sign $0.998 - 0.0455i$
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  + (−0.678 + 2.53i)2-s + (−2.48 − 1.43i)4-s + (3.15 − 3.87i)5-s + (0.662 − 2.47i)7-s + (−2.09 + 2.09i)8-s + (7.68 + 10.6i)10-s + (−1.02 − 1.78i)11-s + (−5.98 − 22.3i)13-s + (5.81 + 3.35i)14-s + (−9.62 − 16.6i)16-s + (14.0 + 14.0i)17-s − 7.13i·19-s + (−13.4 + 5.12i)20-s + (5.21 − 1.39i)22-s + (−4.60 − 17.1i)23-s + ⋯
L(s)  = 1  + (−0.339 + 1.26i)2-s + (−0.621 − 0.358i)4-s + (0.630 − 0.775i)5-s + (0.0946 − 0.353i)7-s + (−0.261 + 0.261i)8-s + (0.768 + 1.06i)10-s + (−0.0935 − 0.162i)11-s + (−0.460 − 1.71i)13-s + (0.415 + 0.239i)14-s + (−0.601 − 1.04i)16-s + (0.826 + 0.826i)17-s − 0.375i·19-s + (−0.670 + 0.256i)20-s + (0.236 − 0.0634i)22-s + (−0.200 − 0.746i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0455i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.998 - 0.0455i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.998 - 0.0455i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.38442 + 0.0315619i\)
\(L(\frac12)\) \(\approx\) \(1.38442 + 0.0315619i\)
\(L(2)\) 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 \( 1 + (-3.15 + 3.87i)T \)
good2 \( 1 + (0.678 - 2.53i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (-0.662 + 2.47i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (1.02 + 1.78i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (5.98 + 22.3i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (-14.0 - 14.0i)T + 289iT^{2} \)
19 \( 1 + 7.13iT - 361T^{2} \)
23 \( 1 + (4.60 + 17.1i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (-31.6 + 18.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.0 + 19.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (39.3 + 39.3i)T + 1.36e3iT^{2} \)
41 \( 1 + (-7.34 + 12.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-64.1 - 17.1i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (20.6 - 76.8i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (21.4 - 21.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (31.9 + 18.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (105. - 28.1i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 102.T + 5.04e3T^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 5.32e3iT^{2} \)
79 \( 1 + (-63.6 + 36.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (77.4 + 20.7i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 73.0iT - 7.92e3T^{2} \)
97 \( 1 + (-28.6 + 107. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−10.72984688021048682854853357109, −9.964219950044972768666621314902, −8.939846454978564011118250738236, −8.085266870805814651508988563517, −7.53958619932608635579282961442, −6.13655493484251970288089220313, −5.63491957932190169725060871300, −4.54936847175368645388956837843, −2.71258098063012543380454323234, −0.68956417899213468693929622208, 1.55400048936121879775901544635, 2.51692064487750074930670690624, 3.57019724144541015223739782416, 5.06197652469004409862748885071, 6.40883284198390165162190129869, 7.19410681929915030746919549748, 8.731792406371012816020202756850, 9.594181264253505196540279257964, 10.09929862728840274996082442102, 11.01701082479573460094102914971

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