Properties

Label 2-405-45.29-c0-0-0
Degree $2$
Conductor $405$
Sign $-0.342 - 0.939i$
Analytic cond. $0.202121$
Root an. cond. $0.449579$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s − 8-s − 0.999·10-s + (0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (0.5 − 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (1 − 1.73i)47-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)5-s − 8-s − 0.999·10-s + (0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)34-s + (0.5 − 0.866i)38-s + (−0.500 − 0.866i)40-s + 0.999·46-s + (1 − 1.73i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(0.202121\)
Root analytic conductor: \(0.449579\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7040183938\)
\(L(\frac12)\) \(\approx\) \(0.7040183938\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−11.75592060411583780117775129241, −10.58023505175654308698770872629, −9.926452438481325881254636165547, −8.803308771221547068513593430479, −8.019681034380590845444153519589, −6.99376370935908539309254717851, −6.38617212887298080680866494517, −5.40367825166858019471337999689, −3.64897770546299677739154530805, −2.41658163365104649263169340771, 1.30758115796568570267697600701, 2.56436313690679554797468672672, 4.09584441689670257259431011088, 5.50020971905860274164625045316, 6.23157570783589567837452934809, 7.79060459310188126284096061338, 8.761493075740315147376689294646, 9.573162261905505329327860118547, 10.17139498500539334332012494454, 11.12919523441986864015023418405

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