Properties

Label 2-405-5.4-c3-0-65
Degree $2$
Conductor $405$
Sign $-0.522 - 0.852i$
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  − 3.08i·2-s − 1.53·4-s + (−5.83 − 9.53i)5-s − 31.3i·7-s − 19.9i·8-s + (−29.4 + 18.0i)10-s − 19.1·11-s + 20.9i·13-s − 96.6·14-s − 73.9·16-s − 6.19i·17-s + 96.6·19-s + (8.94 + 14.6i)20-s + 59.2i·22-s − 162. i·23-s + ⋯
L(s)  = 1  − 1.09i·2-s − 0.191·4-s + (−0.522 − 0.852i)5-s − 1.69i·7-s − 0.882i·8-s + (−0.930 + 0.569i)10-s − 0.526·11-s + 0.447i·13-s − 1.84·14-s − 1.15·16-s − 0.0883i·17-s + 1.16·19-s + (0.0999 + 0.163i)20-s + 0.574i·22-s − 1.47i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.367666274\)
\(L(\frac12)\) \(\approx\) \(1.367666274\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.83 + 9.53i)T \)
good2 \( 1 + 3.08iT - 8T^{2} \)
7 \( 1 + 31.3iT - 343T^{2} \)
11 \( 1 + 19.1T + 1.33e3T^{2} \)
13 \( 1 - 20.9iT - 2.19e3T^{2} \)
17 \( 1 + 6.19iT - 4.91e3T^{2} \)
19 \( 1 - 96.6T + 6.85e3T^{2} \)
23 \( 1 + 162. iT - 1.21e4T^{2} \)
29 \( 1 + 7.64T + 2.43e4T^{2} \)
31 \( 1 - 225.T + 2.97e4T^{2} \)
37 \( 1 - 155. iT - 5.06e4T^{2} \)
41 \( 1 + 315.T + 6.89e4T^{2} \)
43 \( 1 - 192. iT - 7.95e4T^{2} \)
47 \( 1 - 318. iT - 1.03e5T^{2} \)
53 \( 1 + 277. iT - 1.48e5T^{2} \)
59 \( 1 - 429.T + 2.05e5T^{2} \)
61 \( 1 + 89.9T + 2.26e5T^{2} \)
67 \( 1 - 583. iT - 3.00e5T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 + 259. iT - 3.89e5T^{2} \)
79 \( 1 + 124.T + 4.93e5T^{2} \)
83 \( 1 + 36.7iT - 5.71e5T^{2} \)
89 \( 1 + 333.T + 7.04e5T^{2} \)
97 \( 1 + 1.02e3iT - 9.12e5T^{2} \)
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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

−10.27011249904469684809688523750, −9.759250029084080314064942322460, −8.440038876269831237581602116215, −7.48177916375458848719527316796, −6.62047416551655501074017380918, −4.80855055922249534574358489618, −4.10590615807670455317372551911, −3.02083594635107758888754971675, −1.33906801171567710063387281682, −0.47389799149398504740708560093, 2.29091835637323874019171129800, 3.26788200183086249044650671471, 5.19317539668540732863389724180, 5.74403668773748917870507784261, 6.76827223768931724282133351549, 7.70358809745135241593402372049, 8.336162472539263917493984273759, 9.389585267744648505243833171978, 10.54709815765505532423113283784, 11.73952509126665834060376551684

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