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Properties

Label 2-405-9.2-c2-0-28
Degree $2$
Conductor $405$
Sign $0.766 + 0.642i$
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  + (1.93 + 1.11i)2-s + (0.5 + 0.866i)4-s + (1.93 − 1.11i)5-s + (3 − 5.19i)7-s − 6.70i·8-s + 5.00·10-s + (−3.87 − 2.23i)11-s + (−8 − 13.8i)13-s + (11.6 − 6.70i)14-s + (9.5 − 16.4i)16-s + 4.47i·17-s − 2·19-s + (1.93 + 1.11i)20-s + (−5 − 8.66i)22-s + (11.6 − 6.70i)23-s + ⋯
L(s)  = 1  + (0.968 + 0.559i)2-s + (0.125 + 0.216i)4-s + (0.387 − 0.223i)5-s + (0.428 − 0.742i)7-s − 0.838i·8-s + 0.500·10-s + (−0.352 − 0.203i)11-s + (−0.615 − 1.06i)13-s + (0.829 − 0.479i)14-s + (0.593 − 1.02i)16-s + 0.263i·17-s − 0.105·19-s + (0.0968 + 0.0559i)20-s + (−0.227 − 0.393i)22-s + (0.505 − 0.291i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.61165 - 0.950565i\)
\(L(\frac12)\) \(\approx\) \(2.61165 - 0.950565i\)
\(L(2)\) 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 \( 1 + (-1.93 + 1.11i)T \)
good2 \( 1 + (-1.93 - 1.11i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (-3 + 5.19i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (3.87 + 2.23i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (8 + 13.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 4.47iT - 289T^{2} \)
19 \( 1 + 2T + 361T^{2} \)
23 \( 1 + (-11.6 + 6.70i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-27.1 - 15.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-9 - 15.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (-54.2 + 31.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8 - 13.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-42.6 - 24.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 4.47iT - 2.80e3T^{2} \)
59 \( 1 + (-3.87 + 2.23i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (41 - 71.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12 + 20.7i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 + 74T + 5.32e3T^{2} \)
79 \( 1 + (69 - 119. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-81.3 - 46.9i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 + (-83 + 143. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.71102812448933969122661947668, −10.28308228694771716374505058761, −9.088776437609444341733987899843, −7.88486131985701260510249919584, −7.04291992831932314593912657488, −5.94829811397003945995900373017, −5.10539499472028878433399908527, −4.30170947849574776750268643767, −2.94248267544347463772944861776, −0.924880755090013579439579902480, 2.01565948572333217423528775886, 2.83592150039026983523260899663, 4.30114065838001335073272422810, 5.09173139067592330260683461908, 6.06975599757454383033116494567, 7.33886291828441924473253920519, 8.482699100305424632749467593197, 9.374878688843008353424723605096, 10.45965894714322268593532905283, 11.53473971016518917637721004223

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