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Properties

Label 2-405-45.29-c2-0-16
Degree $2$
Conductor $405$
Sign $0.642 - 0.766i$
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 + (2.5 + 4.33i)5-s + 7·8-s + 5·10-s + (−2.5 + 4.33i)16-s + 14·17-s − 22·19-s + (−7.50 + 12.9i)20-s + (17 + 29.4i)23-s + (−12.5 + 21.6i)25-s + (−1 − 1.73i)31-s + (16.5 + 28.5i)32-s + (7 − 12.1i)34-s + (−11 + 19.0i)38-s + ⋯
L(s)  = 1  + (0.250 − 0.433i)2-s + (0.375 + 0.649i)4-s + (0.5 + 0.866i)5-s + 0.875·8-s + 0.5·10-s + (−0.156 + 0.270i)16-s + 0.823·17-s − 1.15·19-s + (−0.375 + 0.649i)20-s + (0.739 + 1.28i)23-s + (−0.500 + 0.866i)25-s + (−0.0322 − 0.0558i)31-s + (0.515 + 0.893i)32-s + (0.205 − 0.356i)34-s + (−0.289 + 0.501i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.642 - 0.766i$
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: \((2,\ 405,\ (\ :1),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.13293 + 0.994605i\)
\(L(\frac12)\) \(\approx\) \(2.13293 + 0.994605i\)
\(L(2)\) 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 \( 1 + (-0.5 + 0.866i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + (84.5 - 146. i)T^{2} \)
17 \( 1 - 14T + 289T^{2} \)
19 \( 1 + 22T + 361T^{2} \)
23 \( 1 + (-17 - 29.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7 - 12.1i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 86T + 2.80e3T^{2} \)
59 \( 1 + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-59 + 102. i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 5.32e3T^{2} \)
79 \( 1 + (49 - 84.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-77 + 133. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.17708905180259586583681831667, −10.47538770597676681047998078786, −9.566600354970446415429227173205, −8.305045179061261151664594749917, −7.34223680721777854611560757242, −6.56808034587258515696422234014, −5.36471371817810702952132776628, −3.88401710049670527581232413743, −2.97176544067916999604825432161, −1.83216309759707665518493366196, 0.994379589510945217032105007996, 2.34829681168297189790167771744, 4.29164944346108434254782178622, 5.21181337918442184444969493154, 6.05520022633615213076379536416, 6.94471855151121318315444837593, 8.161177935817064242033359215539, 9.056464400333833658075718421714, 10.10479142141370149833633487312, 10.71561776157807423894264130213

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