Properties

Label 2-405-9.4-c1-0-1
Degree $2$
Conductor $405$
Sign $-0.984 + 0.173i$
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  + (1.36 + 2.36i)2-s + (−2.73 + 4.73i)4-s + (−0.5 + 0.866i)5-s + (0.633 + 1.09i)7-s − 9.46·8-s − 2.73·10-s + (1.13 + 1.96i)11-s + (2.73 − 4.73i)13-s + (−1.73 + 3i)14-s + (−7.46 − 12.9i)16-s + 0.732·17-s − 2.46·19-s + (−2.73 − 4.73i)20-s + (−3.09 + 5.36i)22-s + (−1.73 + 3i)23-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)2-s + (−1.36 + 2.36i)4-s + (−0.223 + 0.387i)5-s + (0.239 + 0.415i)7-s − 3.34·8-s − 0.863·10-s + (0.341 + 0.592i)11-s + (0.757 − 1.31i)13-s + (−0.462 + 0.801i)14-s + (−1.86 − 3.23i)16-s + 0.177·17-s − 0.565·19-s + (−0.610 − 1.05i)20-s + (−0.660 + 1.14i)22-s + (−0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165825 - 1.89539i\)
\(L(\frac12)\) \(\approx\) \(0.165825 - 1.89539i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.13 - 1.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.73 + 4.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.732T + 17T^{2} \)
19 \( 1 + 2.46T + 19T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.59 - 6.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.732T + 37T^{2} \)
41 \( 1 + (1.59 - 2.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.63 - 4.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.26T + 53T^{2} \)
59 \( 1 + (-5.86 + 10.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.267T + 71T^{2} \)
73 \( 1 - 9.66T + 73T^{2} \)
79 \( 1 + (4.26 + 7.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.09 - 7.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + (3.83 + 6.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.18207642515705751637940758412, −11.02205020827738593807648067126, −9.603806174748282933469960518479, −8.428848140622265578115970261404, −7.899804467210816922651734806742, −6.88765769128456567961206306058, −6.04479816052292695329768466713, −5.19779654704415700788058986464, −4.10165376231819603599411643757, −3.04367738719417493586361704995, 1.01053617757719889937601477821, 2.35090648063785735326048693140, 3.90719652756023183783281195154, 4.28105763848237623438696308070, 5.57693200434407207387072893433, 6.59904456475014027428731226369, 8.488341614347387466734063390495, 9.180599278617522443174533777525, 10.30567940076423927479461632115, 10.94902446248956599800103503468

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