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Properties

Label 2-405-9.4-c3-0-45
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  + (−0.0745 − 0.129i)2-s + (3.98 − 6.90i)4-s + (2.5 − 4.33i)5-s + (−10.0 − 17.4i)7-s − 2.38·8-s − 0.745·10-s + (−4.94 − 8.56i)11-s + (5.91 − 10.2i)13-s + (−1.50 + 2.60i)14-s + (−31.7 − 54.9i)16-s + 6.09·17-s − 62.6·19-s + (−19.9 − 34.5i)20-s + (−0.737 + 1.27i)22-s + (6.03 − 10.4i)23-s + ⋯
L(s)  = 1  + (−0.0263 − 0.0456i)2-s + (0.498 − 0.863i)4-s + (0.223 − 0.387i)5-s + (−0.543 − 0.941i)7-s − 0.105·8-s − 0.0235·10-s + (−0.135 − 0.234i)11-s + (0.126 − 0.218i)13-s + (−0.0286 + 0.0496i)14-s + (−0.495 − 0.858i)16-s + 0.0869·17-s − 0.756·19-s + (−0.222 − 0.386i)20-s + (−0.00715 + 0.0123i)22-s + (0.0547 − 0.0948i)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: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.284004844\)
\(L(\frac12)\) \(\approx\) \(1.284004844\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.0745 + 0.129i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (10.0 + 17.4i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (4.94 + 8.56i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-5.91 + 10.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 6.09T + 4.91e3T^{2} \)
19 \( 1 + 62.6T + 6.85e3T^{2} \)
23 \( 1 + (-6.03 + 10.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.2 - 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (89.0 - 154. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 216.T + 5.06e4T^{2} \)
41 \( 1 + (-205. + 356. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-24.4 - 42.3i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-307. - 532. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 705.T + 1.48e5T^{2} \)
59 \( 1 + (-247. + 428. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (333. + 576. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (138. - 240. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 239.T + 3.57e5T^{2} \)
73 \( 1 + 919.T + 3.89e5T^{2} \)
79 \( 1 + (258. + 447. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (326. + 564. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 543.T + 7.04e5T^{2} \)
97 \( 1 + (566. + 981. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.59709057661571663877394051884, −9.629310552429274747167685313833, −8.727061355130223445262698277064, −7.41262793223463013479571281580, −6.56961387415581487784734015696, −5.66304881643573286889846532690, −4.55665471890886176181569048407, −3.16354796062829355111268732391, −1.63529370538776058716703248346, −0.40301939822957767792427442784, 2.12001311568755274630448511641, 2.97896072664157458857056528642, 4.20279754652611389322887877102, 5.78269680509660644714048463406, 6.54926287810963035039769319425, 7.51606133961214904040198613179, 8.515215563656668417575720284773, 9.340291843595480700801738512338, 10.40343086545828653567112991892, 11.39191875654736630487142603638

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