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Properties

Label 2-405-45.13-c2-0-3
Degree $2$
Conductor $405$
Sign $0.384 - 0.923i$
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  + (−0.570 − 0.152i)2-s + (−3.16 − 1.82i)4-s + (−4.29 − 2.56i)5-s + (−9.62 − 2.58i)7-s + (3.19 + 3.19i)8-s + (2.05 + 2.12i)10-s + (−7.64 − 13.2i)11-s + (12.5 − 3.36i)13-s + (5.10 + 2.94i)14-s + (5.96 + 10.3i)16-s + (5.30 − 5.30i)17-s + 32.8i·19-s + (8.88 + 15.9i)20-s + (2.33 + 8.72i)22-s + (−34.1 + 9.14i)23-s + ⋯
L(s)  = 1  + (−0.285 − 0.0764i)2-s + (−0.790 − 0.456i)4-s + (−0.858 − 0.513i)5-s + (−1.37 − 0.368i)7-s + (0.399 + 0.399i)8-s + (0.205 + 0.212i)10-s + (−0.694 − 1.20i)11-s + (0.965 − 0.258i)13-s + (0.364 + 0.210i)14-s + (0.372 + 0.645i)16-s + (0.312 − 0.312i)17-s + 1.72i·19-s + (0.444 + 0.797i)20-s + (0.106 + 0.396i)22-s + (−1.48 + 0.397i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.384 - 0.923i$
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: \((2,\ 405,\ (\ :1),\ 0.384 - 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.214614 + 0.143172i\)
\(L(\frac12)\) \(\approx\) \(0.214614 + 0.143172i\)
\(L(2)\) 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 \( 1 + (4.29 + 2.56i)T \)
good2 \( 1 + (0.570 + 0.152i)T + (3.46 + 2i)T^{2} \)
7 \( 1 + (9.62 + 2.58i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (7.64 + 13.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.5 + 3.36i)T + (146. - 84.5i)T^{2} \)
17 \( 1 + (-5.30 + 5.30i)T - 289iT^{2} \)
19 \( 1 - 32.8iT - 361T^{2} \)
23 \( 1 + (34.1 - 9.14i)T + (458. - 264.5i)T^{2} \)
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 \( 1 + (13.4 - 50.1i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-11.5 - 3.09i)T + (1.91e3 + 1.10e3i)T^{2} \)
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 \( 1 + (-15.6 - 58.4i)T + (-3.88e3 + 2.24e3i)T^{2} \)
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 \( 1 + (8.23 - 30.7i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 - 68.7iT - 7.92e3T^{2} \)
97 \( 1 + (-14.6 - 3.91i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.03866337136504494381031356659, −10.14948593123364246021351385515, −9.520022203155699789746242216631, −8.207853150260790224791085490108, −8.065206438775682707798882643258, −6.26432145991676776965063819926, −5.55649113678499597277083918799, −4.06855591967361605005960480633, −3.36296014419827057560241796103, −0.933559689667295801269313101326, 0.17100912959300275222710087443, 2.77081176446716806412346886083, 3.79231489197276825050439085979, 4.78023237070048966447651979886, 6.38215021791519815111826764075, 7.17752182694401226010406915792, 8.163074360093868939146332654328, 9.009778951360302299357303419167, 9.917574415003123182134834809749, 10.64856367974130459591802697294

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