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Properties

Label 2-405-45.22-c2-0-36
Degree $2$
Conductor $405$
Sign $-0.998 + 0.0455i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.678 − 2.53i)2-s + (−2.48 − 1.43i)4-s + (−3.15 + 3.87i)5-s + (0.662 − 2.47i)7-s + (2.09 − 2.09i)8-s + (7.68 + 10.6i)10-s + (1.02 + 1.78i)11-s + (−5.98 − 22.3i)13-s + (−5.81 − 3.35i)14-s + (−9.62 − 16.6i)16-s + (−14.0 − 14.0i)17-s − 7.13i·19-s + (13.4 − 5.12i)20-s + (5.21 − 1.39i)22-s + (4.60 + 17.1i)23-s + ⋯
L(s)  = 1  + (0.339 − 1.26i)2-s + (−0.621 − 0.358i)4-s + (−0.630 + 0.775i)5-s + (0.0946 − 0.353i)7-s + (0.261 − 0.261i)8-s + (0.768 + 1.06i)10-s + (0.0935 + 0.162i)11-s + (−0.460 − 1.71i)13-s + (−0.415 − 0.239i)14-s + (−0.601 − 1.04i)16-s + (−0.826 − 0.826i)17-s − 0.375i·19-s + (0.670 − 0.256i)20-s + (0.236 − 0.0634i)22-s + (0.200 + 0.746i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0455i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.998 + 0.0455i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.998 + 0.0455i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0309343 - 1.35689i\)
\(L(\frac12)\) \(\approx\) \(0.0309343 - 1.35689i\)
\(L(2)\) 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 \( 1 + (3.15 - 3.87i)T \)
good2 \( 1 + (-0.678 + 2.53i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (-0.662 + 2.47i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.02 - 1.78i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (5.98 + 22.3i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (14.0 + 14.0i)T + 289iT^{2} \)
19 \( 1 + 7.13iT - 361T^{2} \)
23 \( 1 + (-4.60 - 17.1i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (31.6 - 18.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.0 + 19.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (39.3 + 39.3i)T + 1.36e3iT^{2} \)
41 \( 1 + (7.34 - 12.7i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-64.1 - 17.1i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-20.6 + 76.8i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-21.4 + 21.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (-31.9 - 18.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (105. - 28.1i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 102.T + 5.04e3T^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 5.32e3iT^{2} \)
79 \( 1 + (-63.6 + 36.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-77.4 - 20.7i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 73.0iT - 7.92e3T^{2} \)
97 \( 1 + (-28.6 + 107. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−10.71504807541243901151927844673, −10.18456315429390139022739005156, −9.041538762951621733650426183261, −7.52964891041320924043521565631, −7.15433563449327780474115996545, −5.45741992394724052920647269445, −4.23116541760475856902713730985, −3.27885953419053542157930839496, −2.37977605824215961032029175463, −0.51202783222283250710488390720, 1.90047844328598413741995117891, 4.09155919720995241460292358975, 4.71945749841986405207173234745, 5.86634652780364775796075584230, 6.76974435331776836094393438255, 7.64360585536265904780977116935, 8.651157120899583380416809960615, 9.119049661488338893807698467220, 10.69034067293978637879209885137, 11.65949919888463415392132338235

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