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Properties

Label 2-405-5.4-c3-0-17
Degree $2$
Conductor $405$
Sign $-0.981 - 0.189i$
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  + 2.67i·2-s + 0.844·4-s + (10.9 + 2.11i)5-s + 15.4i·7-s + 23.6i·8-s + (−5.65 + 29.3i)10-s − 44.5·11-s − 28.0i·13-s − 41.2·14-s − 56.5·16-s + 92.6i·17-s − 49.5·19-s + (9.26 + 1.78i)20-s − 119. i·22-s + 0.922i·23-s + ⋯
L(s)  = 1  + 0.945i·2-s + 0.105·4-s + (0.981 + 0.189i)5-s + 0.832i·7-s + 1.04i·8-s + (−0.178 + 0.928i)10-s − 1.22·11-s − 0.597i·13-s − 0.787·14-s − 0.883·16-s + 1.32i·17-s − 0.598·19-s + (0.103 + 0.0199i)20-s − 1.15i·22-s + 0.00836i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.981 - 0.189i$
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: \((2,\ 405,\ (\ :3/2),\ -0.981 - 0.189i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.996159677\)
\(L(\frac12)\) \(\approx\) \(1.996159677\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.9 - 2.11i)T \)
good2 \( 1 - 2.67iT - 8T^{2} \)
7 \( 1 - 15.4iT - 343T^{2} \)
11 \( 1 + 44.5T + 1.33e3T^{2} \)
13 \( 1 + 28.0iT - 2.19e3T^{2} \)
17 \( 1 - 92.6iT - 4.91e3T^{2} \)
19 \( 1 + 49.5T + 6.85e3T^{2} \)
23 \( 1 - 0.922iT - 1.21e4T^{2} \)
29 \( 1 - 189.T + 2.43e4T^{2} \)
31 \( 1 + 299.T + 2.97e4T^{2} \)
37 \( 1 - 57.8iT - 5.06e4T^{2} \)
41 \( 1 - 287.T + 6.89e4T^{2} \)
43 \( 1 - 0.591iT - 7.95e4T^{2} \)
47 \( 1 - 598. iT - 1.03e5T^{2} \)
53 \( 1 + 146. iT - 1.48e5T^{2} \)
59 \( 1 + 193.T + 2.05e5T^{2} \)
61 \( 1 + 566.T + 2.26e5T^{2} \)
67 \( 1 + 355. iT - 3.00e5T^{2} \)
71 \( 1 - 320.T + 3.57e5T^{2} \)
73 \( 1 + 636. iT - 3.89e5T^{2} \)
79 \( 1 - 287.T + 4.93e5T^{2} \)
83 \( 1 + 285. iT - 5.71e5T^{2} \)
89 \( 1 + 331.T + 7.04e5T^{2} \)
97 \( 1 - 1.82e3iT - 9.12e5T^{2} \)
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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

−10.89852936351303922263736817414, −10.51935910913073473891535193514, −9.226697618107799540318942526172, −8.332524844235878061567420152680, −7.55423218741094230722584509022, −6.25841020985030258646095836785, −5.83055964090286696839503076217, −4.93096808225264653423702057004, −2.87852081918714342739435175246, −1.95543857899441385843886476649, 0.60554296402804699711901727378, 1.97499358715995143006589285920, 2.88757649985365380494797716008, 4.29420943862402306536278354394, 5.43386557227759500236611759511, 6.70592482224053286358884656897, 7.46017313910193252768811197197, 8.923640563613158946591424654599, 9.803736231681825960650813946562, 10.50334195278635167327251873337

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