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Properties

Label 2-1620-5.2-c2-0-9
Degree $2$
Conductor $1620$
Sign $0.131 - 0.991i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.87 − 3.16i)5-s + (3.43 + 3.43i)7-s + 7.26·11-s + (−3.06 + 3.06i)13-s + (−4.69 − 4.69i)17-s − 10.1i·19-s + (−29.8 + 29.8i)23-s + (4.96 + 24.5i)25-s − 33.7i·29-s + 23.3·31-s + (−2.42 − 24.1i)35-s + (40.1 + 40.1i)37-s − 10.5·41-s + (−52.7 + 52.7i)43-s + (−17.0 − 17.0i)47-s + ⋯
L(s)  = 1  + (−0.774 − 0.633i)5-s + (0.491 + 0.491i)7-s + 0.660·11-s + (−0.235 + 0.235i)13-s + (−0.276 − 0.276i)17-s − 0.532i·19-s + (−1.29 + 1.29i)23-s + (0.198 + 0.980i)25-s − 1.16i·29-s + 0.753·31-s + (−0.0692 − 0.691i)35-s + (1.08 + 1.08i)37-s − 0.257·41-s + (−1.22 + 1.22i)43-s + (−0.363 − 0.363i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.131 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.182613244\)
\(L(\frac12)\) \(\approx\) \(1.182613244\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (3.87 + 3.16i)T \)
good7 \( 1 + (-3.43 - 3.43i)T + 49iT^{2} \)
11 \( 1 - 7.26T + 121T^{2} \)
13 \( 1 + (3.06 - 3.06i)T - 169iT^{2} \)
17 \( 1 + (4.69 + 4.69i)T + 289iT^{2} \)
19 \( 1 + 10.1iT - 361T^{2} \)
23 \( 1 + (29.8 - 29.8i)T - 529iT^{2} \)
29 \( 1 + 33.7iT - 841T^{2} \)
31 \( 1 - 23.3T + 961T^{2} \)
37 \( 1 + (-40.1 - 40.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 10.5T + 1.68e3T^{2} \)
43 \( 1 + (52.7 - 52.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (17.0 + 17.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-44.6 + 44.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 13.1iT - 3.48e3T^{2} \)
61 \( 1 + 54.2T + 3.72e3T^{2} \)
67 \( 1 + (-46.6 - 46.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 15.1T + 5.04e3T^{2} \)
73 \( 1 + (-31.1 + 31.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 144. iT - 6.24e3T^{2} \)
83 \( 1 + (58.5 - 58.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 124. iT - 7.92e3T^{2} \)
97 \( 1 + (20.8 + 20.8i)T + 9.40e3iT^{2} \)
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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

−9.450381120283886392495794945038, −8.359891731684417699702449908981, −8.082191350723904725631398353858, −7.04874510694579728928233427068, −6.15138697485811215598500446554, −5.12634918079516849767538981468, −4.43242003074530696118845873651, −3.57605732567490768413501698121, −2.26296528288926795409033006606, −1.08518736146014347981745530523, 0.36082288366883408587721244028, 1.81378577804863196927927748589, 3.05110595427597397911964006821, 4.04220321789156301945284412327, 4.58600980198757554697197962222, 5.92591436270832888431534537694, 6.69102907580597018515296318506, 7.47369466202650552267766117293, 8.156945940695033823728387197110, 8.853067975794822981900371229381

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