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Properties

Label 2-405-15.14-c4-0-56
Degree $2$
Conductor $405$
Sign $0.727 - 0.685i$
Analytic cond. $41.8648$
Root an. cond. $6.47030$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.74·2-s + 17.0·4-s + (17.1 + 18.1i)5-s − 47.8i·7-s + 5.92·8-s + (98.5 + 104. i)10-s + 84.5i·11-s + 223. i·13-s − 274. i·14-s − 238.·16-s + 434.·17-s + 378.·19-s + (291. + 309. i)20-s + 486. i·22-s + 653.·23-s + ⋯
L(s)  = 1  + 1.43·2-s + 1.06·4-s + (0.685 + 0.727i)5-s − 0.975i·7-s + 0.0925·8-s + (0.985 + 1.04i)10-s + 0.698i·11-s + 1.32i·13-s − 1.40i·14-s − 0.931·16-s + 1.50·17-s + 1.04·19-s + (0.729 + 0.774i)20-s + 1.00i·22-s + 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(41.8648\)
Root analytic conductor: \(6.47030\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :2),\ 0.727 - 0.685i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.144990739\)
\(L(\frac12)\) \(\approx\) \(5.144990739\)
\(L(3)\) 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 \( 1 + (-17.1 - 18.1i)T \)
good2 \( 1 - 5.74T + 16T^{2} \)
7 \( 1 + 47.8iT - 2.40e3T^{2} \)
11 \( 1 - 84.5iT - 1.46e4T^{2} \)
13 \( 1 - 223. iT - 2.85e4T^{2} \)
17 \( 1 - 434.T + 8.35e4T^{2} \)
19 \( 1 - 378.T + 1.30e5T^{2} \)
23 \( 1 - 653.T + 2.79e5T^{2} \)
29 \( 1 - 496. iT - 7.07e5T^{2} \)
31 \( 1 + 302.T + 9.23e5T^{2} \)
37 \( 1 - 55.6iT - 1.87e6T^{2} \)
41 \( 1 - 475. iT - 2.82e6T^{2} \)
43 \( 1 + 816. iT - 3.41e6T^{2} \)
47 \( 1 - 870.T + 4.87e6T^{2} \)
53 \( 1 - 2.33e3T + 7.89e6T^{2} \)
59 \( 1 + 1.17e3iT - 1.21e7T^{2} \)
61 \( 1 + 6.91e3T + 1.38e7T^{2} \)
67 \( 1 + 3.87e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.82e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.44e3iT - 2.83e7T^{2} \)
79 \( 1 - 4.89e3T + 3.89e7T^{2} \)
83 \( 1 + 6.51e3T + 4.74e7T^{2} \)
89 \( 1 + 1.38e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.34e4iT - 8.85e7T^{2} \)
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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

−10.95037271895704958236339924178, −9.974439870198546244171952857945, −9.180270353506218345431655413738, −7.29630798967004754254928529680, −6.94326180525681834924646615672, −5.77204106107281493880941759494, −4.87482214321701846722746060205, −3.80042622805243307246448238870, −2.90277962147711577927919330233, −1.47374698383932544201738601912, 0.958135840397216175736042765768, 2.65238143275712307851844951923, 3.42408628716088421508273747418, 4.99807809707831622068703068096, 5.54747288153871789827443689478, 6.05065579689283268784828039369, 7.64598277934622219793535137086, 8.763306328188504676554654598525, 9.558592971884519989415376450315, 10.72694331458324389243654388354

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