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Properties

Label 2-405-45.7-c2-0-0
Degree $2$
Conductor $405$
Sign $-0.829 - 0.558i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.78 − 0.747i)2-s + (3.75 − 2.16i)4-s + (−4.50 − 2.16i)5-s + (−11.5 + 3.08i)7-s + (0.689 − 0.689i)8-s + (−14.1 − 2.67i)10-s + (−6.14 + 10.6i)11-s + (−0.906 − 0.242i)13-s + (−29.7 + 17.2i)14-s + (−7.26 + 12.5i)16-s + (−7.47 − 7.47i)17-s − 25.0i·19-s + (−21.6 + 1.63i)20-s + (−9.19 + 34.3i)22-s + (24.7 + 6.62i)23-s + ⋯
L(s)  = 1  + (1.39 − 0.373i)2-s + (0.939 − 0.542i)4-s + (−0.901 − 0.433i)5-s + (−1.64 + 0.440i)7-s + (0.0861 − 0.0861i)8-s + (−1.41 − 0.267i)10-s + (−0.559 + 0.968i)11-s + (−0.0697 − 0.0186i)13-s + (−2.12 + 1.22i)14-s + (−0.454 + 0.786i)16-s + (−0.439 − 0.439i)17-s − 1.31i·19-s + (−1.08 + 0.0817i)20-s + (−0.417 + 1.55i)22-s + (1.07 + 0.288i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.829 - 0.558i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.829 - 0.558i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0626687 + 0.205256i\)
\(L(\frac12)\) \(\approx\) \(0.0626687 + 0.205256i\)
\(L(2)\) 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 \( 1 + (4.50 + 2.16i)T \)
good2 \( 1 + (-2.78 + 0.747i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (11.5 - 3.08i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (6.14 - 10.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (0.906 + 0.242i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (7.47 + 7.47i)T + 289iT^{2} \)
19 \( 1 + 25.0iT - 361T^{2} \)
23 \( 1 + (-24.7 - 6.62i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (2.93 + 1.69i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.4 + 25.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (2.59 + 2.59i)T + 1.36e3iT^{2} \)
41 \( 1 + (11.6 + 20.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (4.25 + 15.8i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (66.9 - 17.9i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (27.6 - 27.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (75.9 - 43.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.5 + 68.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.5 - 61.8i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 88.7T + 5.04e3T^{2} \)
73 \( 1 + (-12.8 + 12.8i)T - 5.32e3iT^{2} \)
79 \( 1 + (26.2 + 15.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-39.3 - 146. i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + (18.4 - 4.95i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.70682200534088689072468947866, −10.89870055467059034805691275759, −9.561957188905374094551348493113, −8.868047370758516595890848933182, −7.35031335373440368406416151722, −6.54638344209480824424075523933, −5.29138017022722278376608428595, −4.52177765513398565673221958932, −3.39439162963040190652115858622, −2.56474547718957552508914656164, 0.05441161109473012244780301700, 3.22238189631527614893735568751, 3.39372773539478284099935884121, 4.65497386759293276049232668572, 5.97880744682645933067261362612, 6.59325901051199658993659253333, 7.47319555134479232960123872363, 8.686239681762881555321540710935, 9.980863585950721462813115239057, 10.84564534256654275619307164797

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