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Properties

Label 2-1620-9.7-c1-0-1
Degree $2$
Conductor $1620$
Sign $-0.766 - 0.642i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
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  + (−0.5 − 0.866i)5-s + (−1 + 1.73i)7-s + (1.5 − 2.59i)11-s + (2 + 3.46i)13-s − 6·17-s − 7·19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + (1.5 − 2.59i)29-s + (−2.5 − 4.33i)31-s + 1.99·35-s − 4·37-s + (1.5 + 2.59i)41-s + (−4 + 6.92i)43-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s + (0.452 − 0.783i)11-s + (0.554 + 0.960i)13-s − 1.45·17-s − 1.60·19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + (0.278 − 0.482i)29-s + (−0.449 − 0.777i)31-s + 0.338·35-s − 0.657·37-s + (0.234 + 0.405i)41-s + (−0.609 + 1.05i)43-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5263619472\)
\(L(\frac12)\) \(\approx\) \(0.5263619472\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.388187937159574100785106310293, −8.863952412991621595377410101591, −8.453013152085747049787730787258, −7.22126982589462017211327122472, −6.34169824154283675164755073432, −5.85606818802726019282396319318, −4.55282514854067970543481622296, −3.94269624297827293749372943913, −2.72049172470732676531179563086, −1.57282922326437926704894201169, 0.19473956684164253545383490904, 1.88065427957924262752637930011, 3.05430964810839070393835231660, 4.06077036208961075527255749196, 4.71624765719265718780979567555, 6.01723067690882447761689464330, 6.86970356997499493542957910144, 7.17175480690019950376715801518, 8.583795628938584984925678646112, 8.782409712783191266003475107892

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