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Properties

Label 2-405-9.7-c3-0-27
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  + (1.03 − 1.79i)2-s + (1.85 + 3.20i)4-s + (2.5 + 4.33i)5-s + (2.33 − 4.03i)7-s + 24.2·8-s + 10.3·10-s + (4.44 − 7.70i)11-s + (−17.3 − 29.9i)13-s + (−4.83 − 8.36i)14-s + (10.3 − 17.9i)16-s − 2.66·17-s + 125.·19-s + (−9.25 + 16.0i)20-s + (−9.21 − 15.9i)22-s + (65.9 + 114. i)23-s + ⋯
L(s)  = 1  + (0.366 − 0.634i)2-s + (0.231 + 0.400i)4-s + (0.223 + 0.387i)5-s + (0.125 − 0.217i)7-s + 1.07·8-s + 0.327·10-s + (0.121 − 0.211i)11-s + (−0.369 − 0.639i)13-s + (−0.0922 − 0.159i)14-s + (0.161 − 0.279i)16-s − 0.0380·17-s + 1.51·19-s + (−0.103 + 0.179i)20-s + (−0.0893 − 0.154i)22-s + (0.598 + 1.03i)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

\(L(2)\) \(\approx\) \(2.951247267\)
\(L(\frac12)\) \(\approx\) \(2.951247267\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.03 + 1.79i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-2.33 + 4.03i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-4.44 + 7.70i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (17.3 + 29.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 2.66T + 4.91e3T^{2} \)
19 \( 1 - 125.T + 6.85e3T^{2} \)
23 \( 1 + (-65.9 - 114. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (35.6 - 61.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (6.71 + 11.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 283.T + 5.06e4T^{2} \)
41 \( 1 + (-191. - 332. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-169. + 293. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-39.1 + 67.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 254.T + 1.48e5T^{2} \)
59 \( 1 + (16.4 + 28.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (93.4 - 161. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (203. + 352. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 966.T + 3.57e5T^{2} \)
73 \( 1 - 276.T + 3.89e5T^{2} \)
79 \( 1 + (-573. + 993. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-89.0 + 154. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 806.T + 7.04e5T^{2} \)
97 \( 1 + (-619. + 1.07e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.02730891251548005265507786692, −10.11001155048993771106669043847, −9.178709958669912024019676161458, −7.71556200451895236884270395684, −7.33619042651234256941809076674, −5.91786174308685374402658989692, −4.77913439123440446949297423584, −3.49716600881075843651669355473, −2.72362677088017302174826368636, −1.24696411500997427366452586100, 1.08104208751484398891722164523, 2.44628217515099246312006631128, 4.27691285796135177908891566328, 5.14966213343558166725806554805, 6.02634854433963013075911654354, 7.01291955630670132642199641090, 7.83243528314280821531526304512, 9.122553156446914591252019929131, 9.824480081099856934081459019777, 10.90127850501919766349736256629

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