Properties

Label 2-1620-9.7-c1-0-9
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: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.864991599\)
\(L(\frac12)\) \(\approx\) \(1.864991599\)
\(L(\frac{3}{2})\) 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

−9.117103165356682067205774154988, −8.707507576078659200805731718439, −7.64949770793006575000847689344, −6.88927875059497253503524900796, −6.03255574087117669283223451061, −5.39609020589636824390682936124, −4.01050746590354684390878290993, −3.46317722579935583968584931004, −2.16232661081501294189568825430, −0.817107985060873369225355709621, 1.29058377336987598391147609536, 2.28209736958848734266206952127, 3.56683067622703208294517785351, 4.68973545747129347315240438732, 5.13815789471283498099003027906, 6.43878483426655247084553947299, 6.93452850895556673355372459868, 7.948433025806952808223705155257, 8.896083244660351379071625984535, 9.275862770202948730539458817071

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