Properties

Label 2-1620-1.1-c3-0-40
Degree $2$
Conductor $1620$
Sign $-1$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 6.21·7-s − 28.3·11-s − 8.60·13-s − 90.1·17-s + 114.·19-s − 48.4·23-s + 25·25-s + 305.·29-s + 93.5·31-s + 31.0·35-s − 282.·37-s − 28.7·41-s − 354.·43-s − 522.·47-s − 304.·49-s − 66.9·53-s − 141.·55-s − 7.63·59-s + 9.41·61-s − 43.0·65-s − 494.·67-s − 560.·71-s + 1.11e3·73-s − 176.·77-s + 1.04e3·79-s + 45.5·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.335·7-s − 0.777·11-s − 0.183·13-s − 1.28·17-s + 1.37·19-s − 0.438·23-s + 0.200·25-s + 1.95·29-s + 0.542·31-s + 0.150·35-s − 1.25·37-s − 0.109·41-s − 1.25·43-s − 1.62·47-s − 0.887·49-s − 0.173·53-s − 0.347·55-s − 0.0168·59-s + 0.0197·61-s − 0.0821·65-s − 0.901·67-s − 0.937·71-s + 1.79·73-s − 0.260·77-s + 1.48·79-s + 0.0601·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-1$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 5T \)
good7 \( 1 - 6.21T + 343T^{2} \)
11 \( 1 + 28.3T + 1.33e3T^{2} \)
13 \( 1 + 8.60T + 2.19e3T^{2} \)
17 \( 1 + 90.1T + 4.91e3T^{2} \)
19 \( 1 - 114.T + 6.85e3T^{2} \)
23 \( 1 + 48.4T + 1.21e4T^{2} \)
29 \( 1 - 305.T + 2.43e4T^{2} \)
31 \( 1 - 93.5T + 2.97e4T^{2} \)
37 \( 1 + 282.T + 5.06e4T^{2} \)
41 \( 1 + 28.7T + 6.89e4T^{2} \)
43 \( 1 + 354.T + 7.95e4T^{2} \)
47 \( 1 + 522.T + 1.03e5T^{2} \)
53 \( 1 + 66.9T + 1.48e5T^{2} \)
59 \( 1 + 7.63T + 2.05e5T^{2} \)
61 \( 1 - 9.41T + 2.26e5T^{2} \)
67 \( 1 + 494.T + 3.00e5T^{2} \)
71 \( 1 + 560.T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 - 1.04e3T + 4.93e5T^{2} \)
83 \( 1 - 45.5T + 5.71e5T^{2} \)
89 \( 1 + 357.T + 7.04e5T^{2} \)
97 \( 1 + 120.T + 9.12e5T^{2} \)
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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

−8.512688786562910822058899111181, −7.992988409909124229167592250982, −6.92698301228756253723847483261, −6.30049976587366383391185463007, −5.11325680993502854980924483048, −4.75744482642598091176716150796, −3.35056355666252147751378211434, −2.45078976799040588997814903216, −1.39229892139978482185168817521, 0, 1.39229892139978482185168817521, 2.45078976799040588997814903216, 3.35056355666252147751378211434, 4.75744482642598091176716150796, 5.11325680993502854980924483048, 6.30049976587366383391185463007, 6.92698301228756253723847483261, 7.992988409909124229167592250982, 8.512688786562910822058899111181

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