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Properties

Label 2-405-135.122-c1-0-10
Degree $2$
Conductor $405$
Sign $0.950 - 0.309i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.42 + 0.996i)2-s + (0.347 + 0.955i)4-s + (0.893 − 2.04i)5-s + (0.429 + 0.920i)7-s + (0.442 − 1.64i)8-s + (3.31 − 2.02i)10-s + (0.759 + 0.904i)11-s + (2.21 + 3.16i)13-s + (−0.306 + 1.73i)14-s + (3.83 − 3.21i)16-s + (−0.144 − 0.540i)17-s + (3.83 − 2.21i)19-s + (2.26 + 0.141i)20-s + (0.178 + 2.04i)22-s + (−4.21 − 1.96i)23-s + ⋯
L(s)  = 1  + (1.00 + 0.704i)2-s + (0.173 + 0.477i)4-s + (0.399 − 0.916i)5-s + (0.162 + 0.347i)7-s + (0.156 − 0.583i)8-s + (1.04 − 0.640i)10-s + (0.228 + 0.272i)11-s + (0.614 + 0.877i)13-s + (−0.0818 + 0.464i)14-s + (0.957 − 0.803i)16-s + (−0.0351 − 0.131i)17-s + (0.880 − 0.508i)19-s + (0.507 + 0.0315i)20-s + (0.0381 + 0.435i)22-s + (−0.879 − 0.410i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.950 - 0.309i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.950 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43315 + 0.385739i\)
\(L(\frac12)\) \(\approx\) \(2.43315 + 0.385739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.893 + 2.04i)T \)
good2 \( 1 + (-1.42 - 0.996i)T + (0.684 + 1.87i)T^{2} \)
7 \( 1 + (-0.429 - 0.920i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.759 - 0.904i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.21 - 3.16i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (0.144 + 0.540i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.83 + 2.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.21 + 1.96i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.44 - 8.17i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (7.51 - 2.73i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (5.30 - 1.42i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.46 + 0.964i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.0660 + 0.754i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (11.1 - 5.19i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-5.40 - 5.40i)T + 53iT^{2} \)
59 \( 1 + (6.20 + 5.20i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-10.5 - 3.83i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.29 - 0.906i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (6.95 + 4.01i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.11 + 1.36i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.62 + 1.16i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.172 - 0.245i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (-5.41 - 9.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.10 + 0.446i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.67623899831239725618511177197, −10.32042457674265716904268467821, −9.294442058448810293854361725008, −8.607605113169337508657040254634, −7.24969071396843786623761284664, −6.37884616480695090877274878812, −5.36634265316341200226625406954, −4.74077229322157210929371074742, −3.59435898151608738190427732832, −1.58931714460408427420335443553, 1.89983781530157592517414644996, 3.24609331978595731352070488693, 3.89518610739427085197203670996, 5.39323359186809012562702199331, 6.09779029196100950927691354513, 7.45802843856880064941075112228, 8.331713301237553719028556459554, 9.826319510188262037294735469502, 10.52868296658113113769699625647, 11.37290601840743096298218035791

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