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Properties

Label 2-1620-9.4-c1-0-2
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 + (1.36 + 2.36i)7-s + (0.866 + 1.5i)11-s + (−2.73 + 4.73i)13-s + 4.73·17-s − 4.46·19-s + (1.73 − 3i)23-s + (−0.499 − 0.866i)25-s + (−3.86 − 6.69i)29-s + (−2.96 + 5.13i)31-s − 2.73·35-s − 6.19·37-s + (−5.59 + 9.69i)41-s + (−1.63 − 2.83i)43-s + (−0.633 − 1.09i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.516 + 0.894i)7-s + (0.261 + 0.452i)11-s + (−0.757 + 1.31i)13-s + 1.14·17-s − 1.02·19-s + (0.361 − 0.625i)23-s + (−0.0999 − 0.173i)25-s + (−0.717 − 1.24i)29-s + (−0.532 + 0.922i)31-s − 0.461·35-s − 1.01·37-s + (−0.874 + 1.51i)41-s + (−0.249 − 0.431i)43-s + (−0.0924 − 0.160i)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

\(L(1)\) \(\approx\) \(1.204202104\)
\(L(\frac12)\) \(\approx\) \(1.204202104\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.73 - 4.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.86 + 6.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.96 - 5.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 + (5.59 - 9.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.63 + 2.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.633 + 1.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.26T + 53T^{2} \)
59 \( 1 + (3.86 - 6.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.19 - 5.53i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 0.196T + 73T^{2} \)
79 \( 1 + (-7.19 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.56 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + (0.366 + 0.633i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.658481382652946378741387521913, −8.843700058052850983277104623774, −8.184709350317208955006408567364, −7.18950251848095663340745154083, −6.59873461651712085522685444979, −5.55052999413787059335797801356, −4.72045635025100401508336567684, −3.83246052561741967921598481315, −2.56060835470871805314430156297, −1.73573661814151542267623526451, 0.45586241715998568903518467537, 1.69408132961571602188841322198, 3.23429752574645418149738603844, 3.94274339970630202709366820004, 5.10862584647076982724371561991, 5.58111159542779789902313269425, 6.89550435391969659932670409428, 7.61274507021519526990620715933, 8.176383333129909123921049612641, 9.058410935713674418508015546845

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