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Properties

Label 2-405-9.7-c3-0-9
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: $\chi_{405} (136, \cdot )$
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

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

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