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Properties

Label 2-405-135.113-c1-0-9
Degree $2$
Conductor $405$
Sign $0.851 + 0.525i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.639 + 0.0559i)2-s + (−1.56 − 0.275i)4-s + (1.26 − 1.84i)5-s + (3.52 + 2.47i)7-s + (−2.22 − 0.595i)8-s + (0.910 − 1.10i)10-s + (0.157 − 0.433i)11-s + (−0.462 − 5.28i)13-s + (2.11 + 1.77i)14-s + (1.59 + 0.580i)16-s + (4.93 − 1.32i)17-s + (1.03 − 0.598i)19-s + (−2.48 + 2.53i)20-s + (0.125 − 0.268i)22-s + (1.57 + 2.25i)23-s + ⋯
L(s)  = 1  + (0.452 + 0.0395i)2-s + (−0.782 − 0.137i)4-s + (0.564 − 0.825i)5-s + (1.33 + 0.933i)7-s + (−0.786 − 0.210i)8-s + (0.287 − 0.350i)10-s + (0.0476 − 0.130i)11-s + (−0.128 − 1.46i)13-s + (0.565 + 0.474i)14-s + (0.399 + 0.145i)16-s + (1.19 − 0.320i)17-s + (0.237 − 0.137i)19-s + (−0.555 + 0.567i)20-s + (0.0266 − 0.0572i)22-s + (0.329 + 0.469i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.851 + 0.525i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.851 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67068 - 0.473991i\)
\(L(\frac12)\) \(\approx\) \(1.67068 - 0.473991i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 + 1.84i)T \)
good2 \( 1 + (-0.639 - 0.0559i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (-3.52 - 2.47i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-0.157 + 0.433i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.462 + 5.28i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-4.93 + 1.32i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.03 + 0.598i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.57 - 2.25i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-0.00462 + 0.00387i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.387 - 2.19i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.94 + 7.25i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.02 - 3.60i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.806 - 1.72i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (6.47 - 9.24i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (3.94 - 3.94i)T - 53iT^{2} \)
59 \( 1 + (-0.388 + 0.141i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.399 - 2.26i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.84 - 0.511i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (0.648 + 0.374i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.97 - 11.1i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.01 + 1.20i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.930 - 10.6i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (6.36 + 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.76 - 3.62i)T + (62.3 - 74.3i)T^{2} \)
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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

−11.34440198517708888693696239506, −10.10737187821510346847605660076, −9.276756007403426032568397513578, −8.452813885615492645409613846581, −7.78202054942514493910029503488, −5.74322721400139655733328843791, −5.41055720376406885396177574405, −4.62031718600914839971054080877, −3.02222430839347785232935074425, −1.24419170051776421428207509081, 1.69946867104208499804346607872, 3.43561131584550999843099193333, 4.45549307171920720720901283284, 5.32968712898257069480646331357, 6.60858646685676129617135592450, 7.58740477950346533572320980897, 8.549204665952725100557818532278, 9.668551508361898667035408548950, 10.40135876114574509945392155774, 11.42633120253261555903737507350

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