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Properties

Label 2-405-9.4-c3-0-38
Degree $2$
Conductor $405$
Sign $0.939 + 0.342i$
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  + (2.60 + 4.50i)2-s + (−9.52 + 16.4i)4-s + (−2.5 + 4.33i)5-s + (−12.2 − 21.1i)7-s − 57.4·8-s − 26.0·10-s + (−14.4 − 25.1i)11-s + (32.6 − 56.6i)13-s + (63.4 − 109. i)14-s + (−73.2 − 126. i)16-s − 68.1·17-s + 104.·19-s + (−47.6 − 82.4i)20-s + (75.3 − 130. i)22-s + (−77.4 + 134. i)23-s + ⋯
L(s)  = 1  + (0.919 + 1.59i)2-s + (−1.19 + 2.06i)4-s + (−0.223 + 0.387i)5-s + (−0.658 − 1.14i)7-s − 2.53·8-s − 0.822·10-s + (−0.397 − 0.688i)11-s + (0.697 − 1.20i)13-s + (1.21 − 2.09i)14-s + (−1.14 − 1.98i)16-s − 0.972·17-s + 1.26·19-s + (−0.532 − 0.922i)20-s + (0.730 − 1.26i)22-s + (−0.701 + 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
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: \((2,\ 405,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8756072511\)
\(L(\frac12)\) \(\approx\) \(0.8756072511\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.60 - 4.50i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (12.2 + 21.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (14.4 + 25.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-32.6 + 56.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 68.1T + 4.91e3T^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 + (77.4 - 134. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (102. + 178. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-9.12 + 15.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 + (97.9 - 169. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (167. + 290. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (2.50 + 4.33i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 319.T + 1.48e5T^{2} \)
59 \( 1 + (-215. + 372. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (297. + 514. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (97.9 - 169. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 425.T + 3.57e5T^{2} \)
73 \( 1 - 929.T + 3.89e5T^{2} \)
79 \( 1 + (12.2 + 21.1i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (272. + 472. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 84.1T + 7.04e5T^{2} \)
97 \( 1 + (413. + 716. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.81844426164035141891002322403, −9.756944080094272382340522223566, −8.416562591837205285775580450276, −7.67526332343942880095191301375, −6.97073589207881097377120362914, −6.03398827116268117935322505160, −5.22898878675605127457248327251, −3.79127065343419629751175203851, −3.34377090671405162206838438883, −0.20917988851622874198767539088, 1.62474088093926047367559464018, 2.60821841377420355361848665918, 3.76166591245201679644262890966, 4.76057891308172857997710867637, 5.64688845676059111291399017992, 6.80355076160951773215539440948, 8.725506296378688709741378879385, 9.251448068262493473307825734451, 10.18817761758887606000441427938, 11.14391754753700368717054264396

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