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Properties

Label 2-405-9.7-c3-0-35
Degree $2$
Conductor $405$
Sign $-0.173 + 0.984i$
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  + (1.29 − 2.23i)2-s + (0.659 + 1.14i)4-s + (−2.5 − 4.33i)5-s + (11.4 − 19.8i)7-s + 24.0·8-s − 12.9·10-s + (−5.54 + 9.59i)11-s + (5.81 + 10.0i)13-s + (−29.5 − 51.2i)14-s + (25.8 − 44.7i)16-s + 10.0·17-s + 117.·19-s + (3.29 − 5.70i)20-s + (14.3 + 24.8i)22-s + (−86.2 − 149. i)23-s + ⋯
L(s)  = 1  + (0.456 − 0.791i)2-s + (0.0824 + 0.142i)4-s + (−0.223 − 0.387i)5-s + (0.618 − 1.07i)7-s + 1.06·8-s − 0.408·10-s + (−0.151 + 0.263i)11-s + (0.124 + 0.215i)13-s + (−0.564 − 0.978i)14-s + (0.404 − 0.699i)16-s + 0.143·17-s + 1.42·19-s + (0.0368 − 0.0638i)20-s + (0.138 + 0.240i)22-s + (−0.781 − 1.35i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 405,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.828139379\)
\(L(\frac12)\) \(\approx\) \(2.828139379\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.29 + 2.23i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-11.4 + 19.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (5.54 - 9.59i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-5.81 - 10.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 10.0T + 4.91e3T^{2} \)
19 \( 1 - 117.T + 6.85e3T^{2} \)
23 \( 1 + (86.2 + 149. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (89.1 - 154. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (70.2 + 121. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 250.T + 5.06e4T^{2} \)
41 \( 1 + (180. + 313. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-180. + 312. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-300. + 519. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 201.T + 1.48e5T^{2} \)
59 \( 1 + (207. + 360. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-27.3 + 47.3i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-265. - 459. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 933.T + 3.57e5T^{2} \)
73 \( 1 + 560.T + 3.89e5T^{2} \)
79 \( 1 + (405. - 702. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (269. - 466. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 686.T + 7.04e5T^{2} \)
97 \( 1 + (357. - 618. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.72066962900113109895471921281, −10.05661849544713952159867129870, −8.720499091392455474217094098698, −7.63775368628411541312660906659, −7.12918901732221681930657069210, −5.38795242752034981059773314225, −4.34034338598726265666252481410, −3.64824043422593475547455410892, −2.15980097699267147441947046555, −0.887283689555789082636028240071, 1.50303581040574459981454260874, 2.98138116172490221368962435327, 4.48291851025284081658966766969, 5.62547380080525757078993569934, 6.00453328877064977625245855454, 7.48224485942857896617528936801, 7.88812488651227874500410778744, 9.213033941651062527001831587427, 10.16742028259377699968957197855, 11.35301909071366526396584880131

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