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Properties

Label 2-405-9.7-c3-0-11
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  + (2.61 − 4.53i)2-s + (−9.69 − 16.7i)4-s + (2.5 + 4.33i)5-s + (−16.5 + 28.5i)7-s − 59.5·8-s + 26.1·10-s + (3.04 − 5.26i)11-s + (32.2 + 55.8i)13-s + (86.3 + 149. i)14-s + (−78.3 + 135. i)16-s + 76.9·17-s − 118.·19-s + (48.4 − 83.9i)20-s + (−15.9 − 27.5i)22-s + (38.8 + 67.2i)23-s + ⋯
L(s)  = 1  + (0.925 − 1.60i)2-s + (−1.21 − 2.09i)4-s + (0.223 + 0.387i)5-s + (−0.891 + 1.54i)7-s − 2.63·8-s + 0.827·10-s + (0.0833 − 0.144i)11-s + (0.687 + 1.19i)13-s + (1.64 + 2.85i)14-s + (−1.22 + 2.12i)16-s + 1.09·17-s − 1.43·19-s + (0.541 − 0.938i)20-s + (−0.154 − 0.267i)22-s + (0.351 + 0.609i)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: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.005860437\)
\(L(\frac12)\) \(\approx\) \(2.005860437\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.61 + 4.53i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (16.5 - 28.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-3.04 + 5.26i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-32.2 - 55.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 76.9T + 4.91e3T^{2} \)
19 \( 1 + 118.T + 6.85e3T^{2} \)
23 \( 1 + (-38.8 - 67.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-31.4 + 54.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-53.4 - 92.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 108.T + 5.06e4T^{2} \)
41 \( 1 + (-71.3 - 123. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (169. - 294. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (299. - 518. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 488.T + 1.48e5T^{2} \)
59 \( 1 + (-121. - 210. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-249. + 432. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-460. - 798. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 60.6T + 3.57e5T^{2} \)
73 \( 1 + 338.T + 3.89e5T^{2} \)
79 \( 1 + (-278. + 481. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (32.4 - 56.1i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 941.T + 7.04e5T^{2} \)
97 \( 1 + (-521. + 902. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.20446273074861696687827725197, −9.968043614660232158843195497119, −9.447787349376115899696504428849, −8.538249961885060133963265890907, −6.38622961608669575556772693707, −5.90863652491493956545996985165, −4.67533609361371024279473447212, −3.44181621634087323618452172529, −2.65764559195259281293225237104, −1.56538808375926424594386531574, 0.49785344509931927622944871879, 3.34605923110453534565544110342, 4.09057709669729085362113269157, 5.18153754826948012835701391874, 6.21929122214589617099281320166, 6.86146558946670722549587037714, 7.84524676164309060633449529051, 8.543405680224243792101626032182, 9.867344719927311248912196622030, 10.70308401166161381011405723868

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