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Properties

Label 2-405-5.4-c3-0-2
Degree $2$
Conductor $405$
Sign $0.930 - 0.367i$
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  − 4.07i·2-s − 8.61·4-s + (−10.3 + 4.10i)5-s − 13.3i·7-s + 2.51i·8-s + (16.7 + 42.3i)10-s − 11.5·11-s − 40.0i·13-s − 54.3·14-s − 58.6·16-s + 93.3i·17-s − 75.1·19-s + (89.5 − 35.3i)20-s + 47.1i·22-s + 142. i·23-s + ⋯
L(s)  = 1  − 1.44i·2-s − 1.07·4-s + (−0.930 + 0.367i)5-s − 0.720i·7-s + 0.110i·8-s + (0.529 + 1.34i)10-s − 0.316·11-s − 0.855i·13-s − 1.03·14-s − 0.917·16-s + 1.33i·17-s − 0.906·19-s + (1.00 − 0.395i)20-s + 0.456i·22-s + 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.930 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.930 - 0.367i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.930 - 0.367i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4356498077\)
\(L(\frac12)\) \(\approx\) \(0.4356498077\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.3 - 4.10i)T \)
good2 \( 1 + 4.07iT - 8T^{2} \)
7 \( 1 + 13.3iT - 343T^{2} \)
11 \( 1 + 11.5T + 1.33e3T^{2} \)
13 \( 1 + 40.0iT - 2.19e3T^{2} \)
17 \( 1 - 93.3iT - 4.91e3T^{2} \)
19 \( 1 + 75.1T + 6.85e3T^{2} \)
23 \( 1 - 142. iT - 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 - 248.T + 2.97e4T^{2} \)
37 \( 1 + 82.9iT - 5.06e4T^{2} \)
41 \( 1 - 449.T + 6.89e4T^{2} \)
43 \( 1 - 279. iT - 7.95e4T^{2} \)
47 \( 1 - 33.8iT - 1.03e5T^{2} \)
53 \( 1 - 423. iT - 1.48e5T^{2} \)
59 \( 1 + 615.T + 2.05e5T^{2} \)
61 \( 1 + 502.T + 2.26e5T^{2} \)
67 \( 1 - 57.7iT - 3.00e5T^{2} \)
71 \( 1 - 252.T + 3.57e5T^{2} \)
73 \( 1 - 823. iT - 3.89e5T^{2} \)
79 \( 1 - 205.T + 4.93e5T^{2} \)
83 \( 1 + 646. iT - 5.71e5T^{2} \)
89 \( 1 + 1.32e3T + 7.04e5T^{2} \)
97 \( 1 - 563. iT - 9.12e5T^{2} \)
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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

−10.74969607799237726471966902382, −10.52230343617698479872768263681, −9.390960599244462753821172226129, −8.144881002297941234731917857736, −7.40849402993145244454795809876, −6.08271771059594551035429814604, −4.39565575870305603887057816390, −3.72493981515146253509547145251, −2.70344582231898630744473376887, −1.18017868612074415414107841402, 0.16068947218620211436366387141, 2.50522057766394048244578201069, 4.30588910082560932030883903819, 5.02892215630643236316189616195, 6.17789350472805549915806345716, 7.04503037656855135087232067701, 7.906863641359995508886809804409, 8.699455663880374862406516217541, 9.331615101619455257590550162652, 10.90520282861661887174636825309

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