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Properties

Label 2-405-5.4-c1-0-13
Degree $2$
Conductor $405$
Sign $0.774 + 0.632i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.517i·2-s + 1.73·4-s + (1.73 + 1.41i)5-s − 3.34i·7-s − 1.93i·8-s + (0.732 − 0.896i)10-s − 1.26·11-s − 2.44i·13-s − 1.73·14-s + 2.46·16-s + 5.27i·17-s + 0.732·19-s + (3 + 2.44i)20-s + 0.656i·22-s − 0.517i·23-s + ⋯
L(s)  = 1  − 0.366i·2-s + 0.866·4-s + (0.774 + 0.632i)5-s − 1.26i·7-s − 0.683i·8-s + (0.231 − 0.283i)10-s − 0.382·11-s − 0.679i·13-s − 0.462·14-s + 0.616·16-s + 1.28i·17-s + 0.167·19-s + (0.670 + 0.547i)20-s + 0.139i·22-s − 0.107i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74712 - 0.622665i\)
\(L(\frac12)\) \(\approx\) \(1.74712 - 0.622665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 1.41i)T \)
good2 \( 1 + 0.517iT - 2T^{2} \)
7 \( 1 + 3.34iT - 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 5.27iT - 17T^{2} \)
19 \( 1 - 0.732T + 19T^{2} \)
23 \( 1 + 0.517iT - 23T^{2} \)
29 \( 1 + 0.464T + 29T^{2} \)
31 \( 1 - 0.732T + 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 7.73T + 41T^{2} \)
43 \( 1 + 0.656iT - 43T^{2} \)
47 \( 1 + 2.96iT - 47T^{2} \)
53 \( 1 - 1.03iT - 53T^{2} \)
59 \( 1 - 9.46T + 59T^{2} \)
61 \( 1 + 6.66T + 61T^{2} \)
67 \( 1 - 7.58iT - 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 - 7.96iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−10.81631828059322420162035099194, −10.43580792809852580949375657015, −9.855329642020066056154145328363, −8.243307526199222830001982522747, −7.25673479599362851659694655581, −6.54707955929608127729622950740, −5.54093072296741559012407896773, −3.87736146350070966528789718066, −2.83689218530823484239353468467, −1.46976075957409685173788243196, 1.88394684322367049776889999404, 2.83700895633733327548454161730, 4.91763840541910869190067219532, 5.64991380925280988323133318431, 6.50219553717813507511638928722, 7.58408561070516087620947823555, 8.710500865089784058594105568354, 9.351792481113638999709366464547, 10.39559863896505867937883291921, 11.62807291717503453676594647289

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