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Properties

Label 2-405-9.4-c3-0-29
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  + (−2.05 − 3.56i)2-s + (−4.47 + 7.75i)4-s + (2.5 − 4.33i)5-s + (10.0 + 17.3i)7-s + 3.92·8-s − 20.5·10-s + (−0.839 − 1.45i)11-s + (−5.55 + 9.62i)13-s + (41.2 − 71.3i)14-s + (27.7 + 48.0i)16-s − 8.98·17-s − 50.6·19-s + (22.3 + 38.7i)20-s + (−3.45 + 5.98i)22-s + (107. − 185. i)23-s + ⋯
L(s)  = 1  + (−0.727 − 1.26i)2-s + (−0.559 + 0.969i)4-s + (0.223 − 0.387i)5-s + (0.540 + 0.936i)7-s + 0.173·8-s − 0.651·10-s + (−0.0230 − 0.0398i)11-s + (−0.118 + 0.205i)13-s + (0.786 − 1.36i)14-s + (0.433 + 0.750i)16-s − 0.128·17-s − 0.611·19-s + (0.250 + 0.433i)20-s + (−0.0334 + 0.0579i)22-s + (0.972 − 1.68i)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

\(L(2)\) \(\approx\) \(0.9642778915\)
\(L(\frac12)\) \(\approx\) \(0.9642778915\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.05 + 3.56i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-10.0 - 17.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (0.839 + 1.45i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (5.55 - 9.62i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 8.98T + 4.91e3T^{2} \)
19 \( 1 + 50.6T + 6.85e3T^{2} \)
23 \( 1 + (-107. + 185. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (38.4 + 66.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-136. + 236. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 137.T + 5.06e4T^{2} \)
41 \( 1 + (26.6 - 46.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (147. + 255. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (97.4 + 168. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 450.T + 1.48e5T^{2} \)
59 \( 1 + (-240. + 416. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (337. + 585. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (447. - 774. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 721.T + 3.57e5T^{2} \)
73 \( 1 - 915.T + 3.89e5T^{2} \)
79 \( 1 + (29.8 + 51.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-371. - 643. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.54e3T + 7.04e5T^{2} \)
97 \( 1 + (-560. - 971. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.43786632896350572095707205282, −9.568945110522385202546154351422, −8.692961356168755974252504960321, −8.271849698121668028079708847511, −6.60359505023484964590584579833, −5.42748582457573261694456505992, −4.22724381389204627999261680100, −2.67225932269487929671280332568, −1.90217721410836801737993427478, −0.44705352908189124540736232543, 1.25298031854304650035038111797, 3.21277714380659311118016894630, 4.75320284103788496382408672803, 5.79485250964195612507317485330, 6.92689153019498514573435376534, 7.38293253635330079746228306738, 8.343540775321412017824457499333, 9.220104729396330253594019252108, 10.20419053753417426532446249046, 10.93889002522349475886217694417

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