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Properties

Label 2-405-9.7-c3-0-15
Degree $2$
Conductor $405$
Sign $0.173 - 0.984i$
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  + (0.366 − 0.633i)2-s + (3.73 + 6.46i)4-s + (−2.5 − 4.33i)5-s + (3.46 − 6i)7-s + 11.3·8-s − 3.66·10-s + (−18.7 + 32.4i)11-s + (19.4 + 33.7i)13-s + (−2.53 − 4.39i)14-s + (−25.7 + 44.5i)16-s − 80.9·17-s + 112.·19-s + (18.6 − 32.3i)20-s + (13.7 + 23.7i)22-s + (−6.71 − 11.6i)23-s + ⋯
L(s)  = 1  + (0.129 − 0.224i)2-s + (0.466 + 0.808i)4-s + (−0.223 − 0.387i)5-s + (0.187 − 0.323i)7-s + 0.500·8-s − 0.115·10-s + (−0.513 + 0.890i)11-s + (0.415 + 0.719i)13-s + (−0.0484 − 0.0838i)14-s + (−0.401 + 0.695i)16-s − 1.15·17-s + 1.36·19-s + (0.208 − 0.361i)20-s + (0.133 + 0.230i)22-s + (−0.0608 − 0.105i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.173 - 0.984i$
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: \((2,\ 405,\ (\ :3/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.878567138\)
\(L(\frac12)\) \(\approx\) \(1.878567138\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.366 + 0.633i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-3.46 + 6i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (18.7 - 32.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-19.4 - 33.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 80.9T + 4.91e3T^{2} \)
19 \( 1 - 112.T + 6.85e3T^{2} \)
23 \( 1 + (6.71 + 11.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-21.7 + 37.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-74.8 - 129. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 + (-186. - 322. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (230. - 398. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-107. + 185. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 445.T + 1.48e5T^{2} \)
59 \( 1 + (-200. - 347. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (0.723 - 1.25i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-408. - 706. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 147.T + 3.57e5T^{2} \)
73 \( 1 - 432.T + 3.89e5T^{2} \)
79 \( 1 + (192. - 332. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-630. + 1.09e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 513T + 7.04e5T^{2} \)
97 \( 1 + (-548. + 949. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.28906878482197710473035462247, −10.24792082253639466098691411483, −9.151635966863806415164833379910, −8.170062651296133527567426691680, −7.35795913310593485222133070998, −6.53671508768050018518937398296, −4.89955865502255272083603786556, −4.12383091107829019293933856820, −2.83735845998290954577301072006, −1.53167476644488562601502681461, 0.59654086940795914222944995266, 2.21792792774659061257923252139, 3.45752239275726090823309914421, 5.07246413061253096775066640046, 5.78834962370098925410542698939, 6.76837503013425195667533421007, 7.74202189957773208733922161351, 8.741307382200954612762078789519, 9.870494731795501038752867357680, 10.83182528983817546722339263334

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