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Properties

Label 2-405-9.7-c3-0-20
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  + (−2.5 + 4.33i)2-s + (−8.50 − 14.7i)4-s + (2.5 + 4.33i)5-s + (−4.5 + 7.79i)7-s + 45.0·8-s − 25.0·10-s + (−4 + 6.92i)11-s + (−21.5 − 37.2i)13-s + (−22.5 − 38.9i)14-s + (−44.5 + 77.0i)16-s + 122·17-s − 59·19-s + (42.5 − 73.6i)20-s + (−20 − 34.6i)22-s + (−106.5 − 184. i)23-s + ⋯
L(s)  = 1  + (−0.883 + 1.53i)2-s + (−1.06 − 1.84i)4-s + (0.223 + 0.387i)5-s + (−0.242 + 0.420i)7-s + 1.98·8-s − 0.790·10-s + (−0.109 + 0.189i)11-s + (−0.458 − 0.794i)13-s + (−0.429 − 0.743i)14-s + (−0.695 + 1.20i)16-s + 1.74·17-s − 0.712·19-s + (0.475 − 0.823i)20-s + (−0.193 − 0.335i)22-s + (−0.965 − 1.67i)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

\(L(2)\) \(\approx\) \(0.9737485140\)
\(L(\frac12)\) \(\approx\) \(0.9737485140\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (4.5 - 7.79i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (4 - 6.92i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (21.5 + 37.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 122T + 4.91e3T^{2} \)
19 \( 1 + 59T + 6.85e3T^{2} \)
23 \( 1 + (106.5 + 184. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-112 + 193. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-18 - 31.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 206T + 5.06e4T^{2} \)
41 \( 1 + (-206.5 - 357. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-196 + 339. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (155.5 - 269. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 377T + 1.48e5T^{2} \)
59 \( 1 + (-168.5 - 291. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (20 - 34.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (174 + 301. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 62T + 3.57e5T^{2} \)
73 \( 1 + 1.21e3T + 3.89e5T^{2} \)
79 \( 1 + (-147 + 254. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-267 + 462. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 810T + 7.04e5T^{2} \)
97 \( 1 + (-464 + 803. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.27087771101878282954279051556, −10.13336677412703279302436041059, −9.009453921639664838947535993325, −8.037460931841631288222744420249, −7.51499814036303595034648936932, −6.22013194273450296198752718570, −5.86477058282750707413137012266, −4.56670113856846847007058687048, −2.65320212878399707592651770514, −0.64535816991815658587434041234, 0.849029250873772318234458793979, 1.94667960305857927765196463026, 3.27186059890917882040787797240, 4.22441303149087682504392401537, 5.68070380261179145117892025585, 7.29478940050813931128092727605, 8.187227456481931034869801924744, 9.166319946083738147286743555203, 9.865132684543373014878803904005, 10.44724554396460045355229778703

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