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Properties

Label 2-1620-9.4-c1-0-13
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 + (0.5 + 0.866i)7-s + (−3 − 5.19i)11-s + (0.5 − 0.866i)13-s − 19-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s − 0.999·35-s − 7·37-s + (3 − 5.19i)41-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (3 − 5.19i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s + (−0.904 − 1.56i)11-s + (0.138 − 0.240i)13-s − 0.229·19-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s − 0.169·35-s − 1.15·37-s + (0.468 − 0.811i)41-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (0.428 − 0.742i)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: $\chi_{1620} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4730063947\)
\(L(\frac12)\) \(\approx\) \(0.4730063947\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.949883376218103070743808538599, −8.243469341778876319583538226088, −7.66239061326064564193665794442, −6.63058003134488918092079384721, −5.70196077856203051079475351937, −5.19800022816892829605052492417, −3.74908654342524861089860561508, −3.13249868820268966518145681601, −1.91731612969935062924872538576, −0.17246342800236113239458994900, 1.60095539050902192641329435932, 2.62541077174039872468217678732, 4.04850110351683053181922841006, 4.62823457667509550779019639423, 5.50128365907402003025237188560, 6.59004112898116291910758319679, 7.47101832687215880270867037410, 7.949043532999002027245448905818, 8.961076822919021374393900967243, 9.684165220106962423030180721426

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