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Properties

Label 2-405-9.4-c3-0-24
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  + (0.129 + 0.224i)2-s + (3.96 − 6.87i)4-s + (−2.5 + 4.33i)5-s + (−7.25 − 12.5i)7-s + 4.12·8-s − 1.29·10-s + (24.6 + 42.6i)11-s + (−36.0 + 62.5i)13-s + (1.87 − 3.25i)14-s + (−31.1 − 54.0i)16-s + 118.·17-s + 123.·19-s + (19.8 + 34.3i)20-s + (−6.37 + 11.0i)22-s + (45.7 − 79.2i)23-s + ⋯
L(s)  = 1  + (0.0457 + 0.0792i)2-s + (0.495 − 0.858i)4-s + (−0.223 + 0.387i)5-s + (−0.391 − 0.678i)7-s + 0.182·8-s − 0.0409·10-s + (0.675 + 1.16i)11-s + (−0.769 + 1.33i)13-s + (0.0358 − 0.0620i)14-s + (−0.487 − 0.844i)16-s + 1.68·17-s + 1.48·19-s + (0.221 + 0.384i)20-s + (−0.0617 + 0.107i)22-s + (0.414 − 0.718i)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

\(L(2)\) \(\approx\) \(2.187801119\)
\(L(\frac12)\) \(\approx\) \(2.187801119\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.129 - 0.224i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (7.25 + 12.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.6 - 42.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (36.0 - 62.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 118.T + 4.91e3T^{2} \)
19 \( 1 - 123.T + 6.85e3T^{2} \)
23 \( 1 + (-45.7 + 79.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (87.2 + 151. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-23.1 + 40.0i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 154.T + 5.06e4T^{2} \)
41 \( 1 + (-182. + 315. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (62.8 + 108. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-110. - 191. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 13.6T + 1.48e5T^{2} \)
59 \( 1 + (119. - 207. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-27.2 - 47.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-38.0 + 65.8i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 728.T + 3.57e5T^{2} \)
73 \( 1 + 501.T + 3.89e5T^{2} \)
79 \( 1 + (198. + 344. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-684. - 1.18e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.46e3T + 7.04e5T^{2} \)
97 \( 1 + (167. + 290. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.68352705870099189047850821103, −9.720145849821529471742424055939, −9.505897793234093275121215915497, −7.46106031370537907544368551581, −7.18098958525093012251927232461, −6.15421064117888151131083713102, −4.93888484382848633774736872706, −3.83646226465715178459108914303, −2.30669789818714182908466854968, −0.961002784397783907802598226639, 1.04424829941113261563497570213, 3.04438348941421696905258326635, 3.39375058236226087965472242223, 5.20940107420964925482208548350, 5.99649014382453406562022418469, 7.43127892736214566209358992381, 7.963945637464063248608317251329, 9.013024793919455504614288329741, 9.884354676366408203953961538915, 11.15432951299846003576869340596

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