Properties

Label 2-405-45.4-c1-0-18
Degree $2$
Conductor $405$
Sign $0.376 + 0.926i$
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  + (2.18 − 1.26i)2-s + (2.18 − 3.78i)4-s + (−0.686 + 2.12i)5-s + (3 − 1.73i)7-s − 5.98i·8-s + (1.18 + 5.51i)10-s + (0.686 + 1.18i)11-s + (−3.55 − 2.05i)13-s + (4.37 − 7.57i)14-s + (−3.18 − 5.51i)16-s + 2.52i·17-s − 5.37·19-s + (6.55 + 7.25i)20-s + (3 + 1.73i)22-s + (4.37 + 2.52i)23-s + ⋯
L(s)  = 1  + (1.54 − 0.892i)2-s + (1.09 − 1.89i)4-s + (−0.306 + 0.951i)5-s + (1.13 − 0.654i)7-s − 2.11i·8-s + (0.375 + 1.74i)10-s + (0.206 + 0.358i)11-s + (−0.986 − 0.569i)13-s + (1.16 − 2.02i)14-s + (−0.796 − 1.37i)16-s + 0.612i·17-s − 1.23·19-s + (1.46 + 1.62i)20-s + (0.639 + 0.369i)22-s + (0.911 + 0.526i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61153 - 1.75718i\)
\(L(\frac12)\) \(\approx\) \(2.61153 - 1.75718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.686 - 2.12i)T \)
good2 \( 1 + (-2.18 + 1.26i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.55 + 2.05i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.52iT - 17T^{2} \)
19 \( 1 + 5.37T + 19T^{2} \)
23 \( 1 + (-4.37 - 2.52i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.87 + 4.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.313 - 0.543i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.57iT - 37T^{2} \)
41 \( 1 + (0.686 - 1.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.37 - 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.34iT - 53T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.81 + 3.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.11 - 4.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.11T + 71T^{2} \)
73 \( 1 + 7.57iT - 73T^{2} \)
79 \( 1 + (2.37 + 4.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.62 + 2.67i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-16.1 + 9.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.16029323598938747700680049040, −10.67275585721955677880314459196, −9.868328371214243885466557323574, −8.050693119055952814410540334217, −7.10716522883572909767460595048, −6.05808043814431237370364915415, −4.82317812270950281888451307808, −4.14901285483434908517053176987, −2.97927262985512945510096480213, −1.80804609096258264259078382532, 2.24561715891817798487480735297, 3.88344089987601142274305047382, 4.93480076517137273559271474609, 5.20753414674607111990970021140, 6.53139223598780537474420776547, 7.52647539090682367838501504947, 8.405107337928483516984857010406, 9.198343899365807100968076946426, 11.09123227081544322795577355161, 11.77306148447876850149114551415

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