Properties

Label 2-405-45.29-c2-0-37
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

\(L(\frac{3}{2})\) \(\approx\) \(0.0766113 - 0.164293i\)
\(L(\frac12)\) \(\approx\) \(0.0766113 - 0.164293i\)
\(L(2)\) 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

−10.87219723725438341815567761614, −9.551628077936617551967454490951, −8.476978157658496165256371944774, −8.223913489044351500887777074324, −6.99323642123612158462826833226, −6.16462809648141667525249024065, −4.72510502641948934612071710553, −3.76455212898924814557707386119, −2.21964785285620202033777801894, −0.07796839221838006038521282421, 1.86598458648237018427246833584, 3.02723988755840065197137322263, 4.32778382847608198057906378047, 5.85183929586760904875550215167, 6.61915776285840484272313761167, 7.59633877458315899320980396190, 8.764627998005545138459719887879, 9.792468513161543745655948759513, 10.57600315855219390884594685293, 11.24695729373639647678619778021

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