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Properties

Label 2-405-5.4-c3-0-58
Degree $2$
Conductor $405$
Sign $-0.323 + 0.946i$
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  − 0.785i·2-s + 7.38·4-s + (3.61 − 10.5i)5-s − 20.9i·7-s − 12.0i·8-s + (−8.30 − 2.83i)10-s + 59.3·11-s − 45.9i·13-s − 16.4·14-s + 49.5·16-s + 43.0i·17-s − 140.·19-s + (26.6 − 78.1i)20-s − 46.6i·22-s + 101. i·23-s + ⋯
L(s)  = 1  − 0.277i·2-s + 0.922·4-s + (0.323 − 0.946i)5-s − 1.12i·7-s − 0.533i·8-s + (−0.262 − 0.0897i)10-s + 1.62·11-s − 0.979i·13-s − 0.313·14-s + 0.774·16-s + 0.614i·17-s − 1.69·19-s + (0.298 − 0.873i)20-s − 0.451i·22-s + 0.922i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.323 + 0.946i$
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: \((2,\ 405,\ (\ :3/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.642079126\)
\(L(\frac12)\) \(\approx\) \(2.642079126\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.61 + 10.5i)T \)
good2 \( 1 + 0.785iT - 8T^{2} \)
7 \( 1 + 20.9iT - 343T^{2} \)
11 \( 1 - 59.3T + 1.33e3T^{2} \)
13 \( 1 + 45.9iT - 2.19e3T^{2} \)
17 \( 1 - 43.0iT - 4.91e3T^{2} \)
19 \( 1 + 140.T + 6.85e3T^{2} \)
23 \( 1 - 101. iT - 1.21e4T^{2} \)
29 \( 1 - 12.3T + 2.43e4T^{2} \)
31 \( 1 - 71.9T + 2.97e4T^{2} \)
37 \( 1 - 150. iT - 5.06e4T^{2} \)
41 \( 1 + 51.0T + 6.89e4T^{2} \)
43 \( 1 + 34.4iT - 7.95e4T^{2} \)
47 \( 1 - 117. iT - 1.03e5T^{2} \)
53 \( 1 - 137. iT - 1.48e5T^{2} \)
59 \( 1 - 496.T + 2.05e5T^{2} \)
61 \( 1 + 247.T + 2.26e5T^{2} \)
67 \( 1 + 826. iT - 3.00e5T^{2} \)
71 \( 1 - 260.T + 3.57e5T^{2} \)
73 \( 1 + 372. iT - 3.89e5T^{2} \)
79 \( 1 + 476.T + 4.93e5T^{2} \)
83 \( 1 + 313. iT - 5.71e5T^{2} \)
89 \( 1 + 817.T + 7.04e5T^{2} \)
97 \( 1 + 941. iT - 9.12e5T^{2} \)
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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

−10.54462209083665878592790195543, −9.865632603394504622947629835357, −8.735114240185681831079104917389, −7.77765854211037430982531410943, −6.67506926072200541295568965443, −5.95239340611897264248809314534, −4.42015814631239998972369288524, −3.54300606424514384301981031223, −1.79782346393968862268877824258, −0.883004334132362219806282045639, 1.87376004343772699822657295731, 2.64899443503262398298075914424, 4.11065219811382210905937286041, 5.73900188291986685835976045650, 6.59663294453354824412038101755, 6.88902952868920339097508263908, 8.427337391603414939105266417955, 9.198722744361461762480234048249, 10.27805577148300155987651884041, 11.31721798253562325049742634747

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