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Properties

Label 2-405-45.13-c2-0-41
Degree $2$
Conductor $405$
Sign $-0.976 - 0.216i$
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  + (−0.307 − 0.0822i)2-s + (−3.37 − 1.94i)4-s + (3.87 − 3.16i)5-s + (−4.71 − 1.26i)7-s + (1.77 + 1.77i)8-s + (−1.44 + 0.651i)10-s + (−5.67 − 9.82i)11-s + (7.58 − 2.03i)13-s + (1.34 + 0.775i)14-s + (7.39 + 12.8i)16-s + (−17.3 + 17.3i)17-s − 8.69i·19-s + (−19.2 + 3.11i)20-s + (0.933 + 3.48i)22-s + (−15.7 + 4.22i)23-s + ⋯
L(s)  = 1  + (−0.153 − 0.0411i)2-s + (−0.844 − 0.487i)4-s + (0.774 − 0.632i)5-s + (−0.673 − 0.180i)7-s + (0.221 + 0.221i)8-s + (−0.144 + 0.0651i)10-s + (−0.515 − 0.893i)11-s + (0.583 − 0.156i)13-s + (0.0959 + 0.0553i)14-s + (0.462 + 0.800i)16-s + (−1.02 + 1.02i)17-s − 0.457i·19-s + (−0.962 + 0.155i)20-s + (0.0424 + 0.158i)22-s + (−0.686 + 0.183i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.976 - 0.216i$
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: \((2,\ 405,\ (\ :1),\ -0.976 - 0.216i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0396529 + 0.361658i\)
\(L(\frac12)\) \(\approx\) \(0.0396529 + 0.361658i\)
\(L(2)\) 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 \( 1 + (-3.87 + 3.16i)T \)
good2 \( 1 + (0.307 + 0.0822i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (4.71 + 1.26i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (5.67 + 9.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.58 + 2.03i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (17.3 - 17.3i)T - 289iT^{2} \)
19 \( 1 + 8.69iT - 361T^{2} \)
23 \( 1 + (15.7 - 4.22i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + (30.4 - 17.5i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (5.34 - 9.26i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 + (0.348 - 0.603i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (9.69 - 36.1i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (60.4 + 16.1i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (0.696 + 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 + (34.5 + 19.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.95 + 5.11i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.5 + 61.6i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-21.2 + 12.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-4.81 + 17.9i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (-33.5 - 9.00i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−10.42428454065084684026596939881, −9.589319163900508758706993987892, −8.814373520200205245342488693499, −8.163516794408712381251280078794, −6.43474522554969390248295464042, −5.75432655498199941957377346857, −4.74655436045059052633911038223, −3.49798299850646984600414273768, −1.67668036791352579525836818871, −0.16385126761001890608876670694, 2.18877332901193193389352014644, 3.45115393625661500734754348218, 4.67219442494162932630083170968, 5.85385378250781468611035114837, 6.89229379597589018780588152606, 7.80443012349726384201828099139, 9.044356679927363267748011635629, 9.619110280620622427264811466836, 10.35530915942711173205526956849, 11.50132802629768474853719181423

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