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Properties

Label 2-405-15.14-c4-0-19
Degree $2$
Conductor $405$
Sign $-0.248 - 0.968i$
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  + 1.09·2-s − 14.8·4-s + (24.2 − 6.20i)5-s + 19.5i·7-s − 33.7·8-s + (26.5 − 6.79i)10-s + 115. i·11-s − 183. i·13-s + 21.3i·14-s + 199.·16-s + 89.0·17-s − 52.2·19-s + (−358. + 91.9i)20-s + 126. i·22-s − 464.·23-s + ⋯
L(s)  = 1  + 0.273·2-s − 0.925·4-s + (0.968 − 0.248i)5-s + 0.398i·7-s − 0.526·8-s + (0.265 − 0.0679i)10-s + 0.958i·11-s − 1.08i·13-s + 0.109i·14-s + 0.780·16-s + 0.308·17-s − 0.144·19-s + (−0.896 + 0.229i)20-s + 0.262i·22-s − 0.877·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.248 - 0.968i$
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: \((2,\ 405,\ (\ :2),\ -0.248 - 0.968i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.333499358\)
\(L(\frac12)\) \(\approx\) \(1.333499358\)
\(L(3)\) 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 \( 1 + (-24.2 + 6.20i)T \)
good2 \( 1 - 1.09T + 16T^{2} \)
7 \( 1 - 19.5iT - 2.40e3T^{2} \)
11 \( 1 - 115. iT - 1.46e4T^{2} \)
13 \( 1 + 183. iT - 2.85e4T^{2} \)
17 \( 1 - 89.0T + 8.35e4T^{2} \)
19 \( 1 + 52.2T + 1.30e5T^{2} \)
23 \( 1 + 464.T + 2.79e5T^{2} \)
29 \( 1 - 1.44e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.53e3T + 9.23e5T^{2} \)
37 \( 1 - 641. iT - 1.87e6T^{2} \)
41 \( 1 + 1.38e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.05e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.17e3T + 4.87e6T^{2} \)
53 \( 1 + 64.4T + 7.89e6T^{2} \)
59 \( 1 - 3.27e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.85e3T + 1.38e7T^{2} \)
67 \( 1 - 8.38e3iT - 2.01e7T^{2} \)
71 \( 1 - 4.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.07e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.65e3T + 3.89e7T^{2} \)
83 \( 1 + 3.34e3T + 4.74e7T^{2} \)
89 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.37e4iT - 8.85e7T^{2} \)
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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

−10.61541394881991988755475742795, −9.917286440565659718900277961530, −9.141213518152804938849985534051, −8.356779331832599583310162286117, −7.12842715076002555242462862073, −5.69071526865000579154845776338, −5.31166712857051050583034509574, −4.09855778632220528483309688106, −2.73494105165462451570803640573, −1.31419719056575241298992603833, 0.35868066456952526254996126228, 1.91469305763406005928508636701, 3.43604782190143249487345671023, 4.41017515972464083214763085962, 5.63589294965542082529041321146, 6.26626024755322142867135561243, 7.59854618471779242225554274516, 8.765689888924781995921967829898, 9.432235054725719630636871439741, 10.24563346890408179586962771612

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