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Properties

Label 2-405-9.7-c3-0-31
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  + (1.76 − 3.05i)2-s + (−2.20 − 3.82i)4-s + (−2.5 − 4.33i)5-s + (−12.7 + 22.0i)7-s + 12.6·8-s − 17.6·10-s + (35.6 − 61.7i)11-s + (25.6 + 44.5i)13-s + (44.8 + 77.6i)14-s + (39.9 − 69.1i)16-s − 33.3·17-s + 113.·19-s + (−11.0 + 19.1i)20-s + (−125. − 217. i)22-s + (−40.9 − 70.9i)23-s + ⋯
L(s)  = 1  + (0.622 − 1.07i)2-s + (−0.275 − 0.477i)4-s + (−0.223 − 0.387i)5-s + (−0.686 + 1.18i)7-s + 0.558·8-s − 0.557·10-s + (0.977 − 1.69i)11-s + (0.548 + 0.949i)13-s + (0.855 + 1.48i)14-s + (0.623 − 1.08i)16-s − 0.475·17-s + 1.36·19-s + (−0.123 + 0.213i)20-s + (−1.21 − 2.11i)22-s + (−0.371 − 0.643i)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

\(L(2)\) \(\approx\) \(2.763288948\)
\(L(\frac12)\) \(\approx\) \(2.763288948\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.76 + 3.05i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (12.7 - 22.0i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-35.6 + 61.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-25.6 - 44.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 33.3T + 4.91e3T^{2} \)
19 \( 1 - 113.T + 6.85e3T^{2} \)
23 \( 1 + (40.9 + 70.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-123. + 213. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (111. + 192. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 22.3T + 5.06e4T^{2} \)
41 \( 1 + (-217. - 376. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-118. + 205. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-53.9 + 93.5i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 123.T + 1.48e5T^{2} \)
59 \( 1 + (85.5 + 148. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-39.7 + 68.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-305. - 529. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 511.T + 3.57e5T^{2} \)
73 \( 1 + 410.T + 3.89e5T^{2} \)
79 \( 1 + (-396. + 687. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (135. - 233. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 177.T + 7.04e5T^{2} \)
97 \( 1 + (-440. + 763. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.05896072684238651553853932524, −9.629971972259437566259853266891, −9.009985588308919498185416661888, −8.042153819591032907143387676673, −6.44356807357476632854734062506, −5.68640649122396028675197235691, −4.27926158868649879108515208909, −3.42564324526607843813127527420, −2.38022650239251822366217767131, −0.890137360295800280281725224499, 1.29472192606745831488182616079, 3.43158062442353425444694886287, 4.27922892338312816076940371245, 5.37512536041597489741245992512, 6.62912069454886254751876933617, 7.10874456113919603846184183476, 7.75204321483428634351130375760, 9.290166646308382204843346170797, 10.23066914252154552967532681609, 10.90398021582376262127663176992

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