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Properties

Label 2-405-9.7-c3-0-6
Degree $2$
Conductor $405$
Sign $-0.766 - 0.642i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
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 + (3.5 + 6.06i)4-s + (−2.5 − 4.33i)5-s + (12 − 20.7i)7-s − 15·8-s + 5·10-s + (−26 + 45.0i)11-s + (−11 − 19.0i)13-s + (12 + 20.7i)14-s + (−20.5 + 35.5i)16-s − 14·17-s − 20·19-s + (17.5 − 30.3i)20-s + (−25.9 − 45.0i)22-s + (84 + 145. i)23-s + ⋯
L(s)  = 1  + (−0.176 + 0.306i)2-s + (0.437 + 0.757i)4-s + (−0.223 − 0.387i)5-s + (0.647 − 1.12i)7-s − 0.662·8-s + 0.158·10-s + (−0.712 + 1.23i)11-s + (−0.234 − 0.406i)13-s + (0.229 + 0.396i)14-s + (−0.320 + 0.554i)16-s − 0.199·17-s − 0.241·19-s + (0.195 − 0.338i)20-s + (−0.251 − 0.436i)22-s + (0.761 + 1.31i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.096420320\)
\(L(\frac12)\) \(\approx\) \(1.096420320\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-12 + 20.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (26 - 45.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (11 + 19.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 14T + 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 + (-84 - 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (115 - 199. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-144 - 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 34T + 5.06e4T^{2} \)
41 \( 1 + (61 + 105. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-94 + 162. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (128 - 221. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 338T + 1.48e5T^{2} \)
59 \( 1 + (50 + 86.6i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (371 - 642. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-42 - 72.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 328T + 3.57e5T^{2} \)
73 \( 1 + 38T + 3.89e5T^{2} \)
79 \( 1 + (-120 + 207. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (606 - 1.04e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 330T + 7.04e5T^{2} \)
97 \( 1 + (433 - 749. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.10017514217526254025031349374, −10.41875733219727150687167694112, −9.223814539913341139283393934071, −8.181549506432025174555224532610, −7.40245555299381134868088824153, −6.98395873566930954827219961034, −5.27259794777215097096801024219, −4.34299413072970742879093363113, −3.11898570828131444014454487440, −1.54681215631816145620850469354, 0.36940359264553153231060128000, 2.10226978541862569255523420520, 2.90135624741853459367663268975, 4.70030898147673782840078811960, 5.79321559789760616985500260299, 6.43846183948833700156271911262, 7.86913933031781824076044391248, 8.691881111920917336793162609273, 9.637851764075832224370095580184, 10.69006454888089538715874193701

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