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Properties

Label 2-405-45.13-c2-0-1
Degree $2$
Conductor $405$
Sign $-0.619 - 0.784i$
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  + (0.5 + 0.133i)2-s + (−3.23 − 1.86i)4-s + (−2.5 − 4.33i)5-s + (2.09 + 0.562i)7-s + (−2.83 − 2.83i)8-s + (−0.669 − 2.5i)10-s + (2.09 + 3.63i)11-s + (−14.6 + 3.93i)13-s + (0.973 + 0.562i)14-s + (6.42 + 11.1i)16-s + (−6.70 + 6.70i)17-s − 0.588i·19-s + 18.6i·20-s + (0.562 + 2.09i)22-s + (28.4 − 7.63i)23-s + ⋯
L(s)  = 1  + (0.250 + 0.0669i)2-s + (−0.808 − 0.466i)4-s + (−0.5 − 0.866i)5-s + (0.299 + 0.0803i)7-s + (−0.353 − 0.353i)8-s + (−0.0669 − 0.250i)10-s + (0.190 + 0.330i)11-s + (−1.13 + 0.302i)13-s + (0.0695 + 0.0401i)14-s + (0.401 + 0.695i)16-s + (−0.394 + 0.394i)17-s − 0.0309i·19-s + 0.933i·20-s + (0.0255 + 0.0953i)22-s + (1.23 − 0.331i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.619 - 0.784i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.619 - 0.784i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.104637 + 0.215889i\)
\(L(\frac12)\) \(\approx\) \(0.104637 + 0.215889i\)
\(L(2)\) 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 \( 1 + (2.5 + 4.33i)T \)
good2 \( 1 + (-0.5 - 0.133i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (-2.09 - 0.562i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-2.09 - 3.63i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (14.6 - 3.93i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (6.70 - 6.70i)T - 289iT^{2} \)
19 \( 1 + 0.588iT - 361T^{2} \)
23 \( 1 + (-28.4 + 7.63i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (24.9 - 14.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (28.5 - 49.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (29.8 - 29.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (-23.3 + 40.5i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (14.2 - 53.3i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-26.5 - 7.12i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (47.9 + 47.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (44.1 + 25.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.0 + 26.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (11.0 + 41.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 30.6T + 5.04e3T^{2} \)
73 \( 1 + (5.72 + 5.72i)T + 5.32e3iT^{2} \)
79 \( 1 + (14.8 - 8.58i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-37.5 + 140. i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 173. iT - 7.92e3T^{2} \)
97 \( 1 + (30.9 + 8.28i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.43549318671308651361814304449, −10.41228916980780061695800623377, −9.264836477220739736168219695921, −8.880351809644041271404638072209, −7.73453786948798309441819333349, −6.62543609046065506846090599639, −5.06517993872242692280563626328, −4.85841554234899866840802031731, −3.57743120909753394194440187429, −1.54575483569711302883503612692, 0.099065668301956841509608932043, 2.59265231510924277495717589200, 3.66277185782574290073304972367, 4.65888314410242019860087627878, 5.76623223304469519928320454034, 7.25856867309469429920728193840, 7.72675596258492529563789316288, 8.943181755023219341845814049371, 9.719171891579071167333394274314, 10.93036734812199018685943304634

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