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Properties

Label 2-1620-9.4-c1-0-9
Degree $2$
Conductor $1620$
Sign $0.642 + 0.766i$
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 + (−0.366 − 0.633i)7-s + (0.866 + 1.5i)11-s + (0.732 − 1.26i)13-s − 1.26·17-s + 2.46·19-s + (1.73 − 3i)23-s + (−0.499 − 0.866i)25-s + (2.13 + 3.69i)29-s + (3.96 − 6.86i)31-s − 0.732·35-s + 4.19·37-s + (0.401 − 0.696i)41-s + (−3.36 − 5.83i)43-s + (2.36 + 4.09i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.138 − 0.239i)7-s + (0.261 + 0.452i)11-s + (0.203 − 0.351i)13-s − 0.307·17-s + 0.565·19-s + (0.361 − 0.625i)23-s + (−0.0999 − 0.173i)25-s + (0.396 + 0.686i)29-s + (0.711 − 1.23i)31-s − 0.123·35-s + 0.689·37-s + (0.0627 − 0.108i)41-s + (−0.513 − 0.889i)43-s + (0.345 + 0.597i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.642 + 0.766i$
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: \((2,\ 1620,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.737224623\)
\(L(\frac12)\) \(\approx\) \(1.737224623\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 + 0.633i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.732 + 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.26T + 17T^{2} \)
19 \( 1 - 2.46T + 19T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.13 - 3.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.96 + 6.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 + (-0.401 + 0.696i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.36 + 5.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.36 - 4.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + (-2.13 + 3.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.19 + 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.803T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + (3.19 + 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.56 + 7.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + (-1.36 - 2.36i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.341691004926548433649776213320, −8.515799349846804794505756762891, −7.73597881156011754866160024298, −6.83875699169215445204378109869, −6.08642069652963399029927411949, −5.09143726527632559171599928767, −4.33582107847498803953399312772, −3.27167056965582197922978959604, −2.10526354074546352033617889543, −0.77484711188084090977357244211, 1.21733416237829936893073573232, 2.57229822579732747643466573371, 3.42883372336940212003047841801, 4.50566069069556445041736967464, 5.50738025480898550302218151210, 6.32942161225038323282827338987, 6.97681224019572989391786094236, 7.970957091270820498142075215082, 8.747316199043512435041299928404, 9.537945349303752683151114247292

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