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Properties

Label 2-405-9.4-c3-0-33
Degree $2$
Conductor $405$
Sign $0.984 - 0.173i$
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  + (1.78 + 3.09i)2-s + (−2.38 + 4.13i)4-s + (−2.5 + 4.33i)5-s + (−7.07 − 12.2i)7-s + 11.5·8-s − 17.8·10-s + (−31.8 − 55.1i)11-s + (5.19 − 9.00i)13-s + (25.3 − 43.8i)14-s + (39.6 + 68.7i)16-s + 108.·17-s + 18.6·19-s + (−11.9 − 20.6i)20-s + (113. − 197. i)22-s + (64.4 − 111. i)23-s + ⋯
L(s)  = 1  + (0.631 + 1.09i)2-s + (−0.298 + 0.517i)4-s + (−0.223 + 0.387i)5-s + (−0.382 − 0.662i)7-s + 0.509·8-s − 0.565·10-s + (−0.872 − 1.51i)11-s + (0.110 − 0.192i)13-s + (0.483 − 0.836i)14-s + (0.620 + 1.07i)16-s + 1.54·17-s + 0.224·19-s + (−0.133 − 0.231i)20-s + (1.10 − 1.90i)22-s + (0.583 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.447976288\)
\(L(\frac12)\) \(\approx\) \(2.447976288\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.78 - 3.09i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (7.07 + 12.2i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (31.8 + 55.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-5.19 + 9.00i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 - 18.6T + 6.85e3T^{2} \)
23 \( 1 + (-64.4 + 111. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (41.8 + 72.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-5.23 + 9.06i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 81.8T + 5.06e4T^{2} \)
41 \( 1 + (-153. + 266. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-111. - 192. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (180. + 313. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 562.T + 1.48e5T^{2} \)
59 \( 1 + (231. - 400. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-324. - 562. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (127. - 220. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.09e3T + 3.57e5T^{2} \)
73 \( 1 - 1.03e3T + 3.89e5T^{2} \)
79 \( 1 + (575. + 997. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (262. + 454. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 656.T + 7.04e5T^{2} \)
97 \( 1 + (562. + 974. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.61655494417001296880557520999, −10.28094815694178581238589876322, −8.640920214108262785598153011902, −7.76030876716268352398465606204, −7.08320573699232518674654685562, −5.98077293007292244492135831431, −5.38096494540310057025406028497, −3.99754730565636156907249938209, −3.01053959519038441884542041925, −0.72109176007278871045405310221, 1.39011420775138007015408401250, 2.59624352281687024516706137479, 3.60662631633878096178404450612, 4.82501899586508614595735365556, 5.53274729425361471433551915672, 7.21685397383618803838142423812, 7.946638926353170770169610450062, 9.412072514706420877565612488111, 9.972966852575052982611652923375, 10.99357559421015316651257952121

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