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Properties

Label 2-405-45.43-c2-0-14
Degree $2$
Conductor $405$
Sign $0.676 - 0.736i$
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  + (−0.152 − 0.570i)2-s + (3.16 − 1.82i)4-s + (−4.36 + 2.43i)5-s + (2.58 + 9.62i)7-s + (−3.19 − 3.19i)8-s + (2.05 + 2.12i)10-s + (7.64 − 13.2i)11-s + (−3.36 + 12.5i)13-s + (5.10 − 2.94i)14-s + (5.96 − 10.3i)16-s + (−5.30 + 5.30i)17-s + 32.8i·19-s + (−9.36 + 15.6i)20-s + (−8.72 − 2.33i)22-s + (−9.14 + 34.1i)23-s + ⋯
L(s)  = 1  + (−0.0764 − 0.285i)2-s + (0.790 − 0.456i)4-s + (−0.873 + 0.486i)5-s + (0.368 + 1.37i)7-s + (−0.399 − 0.399i)8-s + (0.205 + 0.212i)10-s + (0.694 − 1.20i)11-s + (−0.258 + 0.965i)13-s + (0.364 − 0.210i)14-s + (0.372 − 0.645i)16-s + (−0.312 + 0.312i)17-s + 1.72i·19-s + (−0.468 + 0.783i)20-s + (−0.396 − 0.106i)22-s + (−0.397 + 1.48i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.676 - 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47797 + 0.649023i\)
\(L(\frac12)\) \(\approx\) \(1.47797 + 0.649023i\)
\(L(2)\) 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 \( 1 + (4.36 - 2.43i)T \)
good2 \( 1 + (0.152 + 0.570i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (-2.58 - 9.62i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-7.64 + 13.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (3.36 - 12.5i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (5.30 - 5.30i)T - 289iT^{2} \)
19 \( 1 - 32.8iT - 361T^{2} \)
23 \( 1 + (9.14 - 34.1i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (-31.4 - 18.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-0.699 - 1.21i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-15.9 + 15.9i)T - 1.36e3iT^{2} \)
41 \( 1 + (-5.14 - 8.91i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-50.1 + 13.4i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-3.09 - 11.5i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-42.9 - 42.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-0.100 + 0.0578i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.12 - 3.68i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (58.4 + 15.6i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 64.5T + 5.04e3T^{2} \)
73 \( 1 + (-12.7 - 12.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (-11.8 - 6.81i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (30.7 - 8.23i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 68.7iT - 7.92e3T^{2} \)
97 \( 1 + (3.91 + 14.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.35750041061523580244701746155, −10.49275751901067585081742337586, −9.310176707937881491440192771970, −8.471346646379636348687123333237, −7.44252721928251805072253459041, −6.27043472715263428109729976117, −5.68566042651765919242765948510, −3.96777049855882171821256224073, −2.85389413175390740380607381195, −1.56384504783948613883444670470, 0.74526544531259080231431653092, 2.62645421707232858783238828184, 4.11986209627041018810779693715, 4.76665718229952010908055357515, 6.59994326318695543811821200083, 7.25257709968816065142863938096, 7.894934327868784068661501953889, 8.855848649404464461910720104128, 10.19191940218502938958045790624, 10.99435228785350765015460448643

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