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Properties

Label 2-405-45.29-c2-0-2
Degree $2$
Conductor $405$
Sign $-0.892 + 0.451i$
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 + 0.866i)2-s + (1.5 + 2.59i)4-s + (4.59 + 1.96i)5-s + (−5.19 − 3i)7-s − 7·8-s + (−4 + 3i)10-s + (−18.1 − 10.5i)11-s + (−12.9 + 7.5i)13-s + (5.19 − 3i)14-s + (−2.5 + 4.33i)16-s − 23·17-s + 14·19-s + (1.79 + 14.8i)20-s + (18.1 − 10.5i)22-s + (−3.5 − 6.06i)23-s + ⋯
L(s)  = 1  + (−0.250 + 0.433i)2-s + (0.375 + 0.649i)4-s + (0.919 + 0.392i)5-s + (−0.742 − 0.428i)7-s − 0.875·8-s + (−0.400 + 0.300i)10-s + (−1.65 − 0.954i)11-s + (−0.999 + 0.576i)13-s + (0.371 − 0.214i)14-s + (−0.156 + 0.270i)16-s − 1.35·17-s + 0.736·19-s + (0.0897 + 0.744i)20-s + (0.826 − 0.477i)22-s + (−0.152 − 0.263i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0942077 - 0.394375i\)
\(L(\frac12)\) \(\approx\) \(0.0942077 - 0.394375i\)
\(L(2)\) 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 \( 1 + (-4.59 - 1.96i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (5.19 + 3i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (18.1 + 10.5i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (12.9 - 7.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 23T + 289T^{2} \)
19 \( 1 - 14T + 361T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (2.59 + 1.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-12.5 - 21.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 54iT - 1.36e3T^{2} \)
41 \( 1 + (-20.7 + 12i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (12.9 + 7.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (24.5 - 42.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 14T + 2.80e3T^{2} \)
59 \( 1 + (25.9 - 15i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (22 - 38.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-57.1 + 33i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 18iT - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 + (-18.5 + 32.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-58 + 100. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 126iT - 7.92e3T^{2} \)
97 \( 1 + (67.5 + 39i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.42615261799197833562374413644, −10.56139565373629960828658484059, −9.729547171539042347593630826298, −8.778375024334617215923072547507, −7.74468841420304782053412539306, −6.84995805397541422783680323933, −6.17297993913756995103645617031, −4.94038357369423227779618455741, −3.18097289480856735503876963621, −2.44574703090298797262429750527, 0.16410200163546636106550756948, 2.12529449979091421560202350156, 2.70903710928992269116259586464, 4.92019504386818500535927591310, 5.58255025240274878446741931027, 6.60537456575548641446587011521, 7.71729515911071523245233415733, 9.131696321188409900584604373112, 9.775543162064087697260250228027, 10.26432006641701311537851003900

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