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Properties

Label 2-1620-12.11-c1-0-11
Degree $2$
Conductor $1620$
Sign $-0.750 - 0.660i$
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  + (−0.499 + 1.32i)2-s + (−1.50 − 1.32i)4-s − i·5-s − 1.20i·7-s + (2.49 − 1.32i)8-s + (1.32 + 0.499i)10-s − 3.04·11-s − 5.07·13-s + (1.59 + 0.603i)14-s + (0.505 + 3.96i)16-s + 2.23i·17-s − 2.53i·19-s + (−1.32 + 1.50i)20-s + (1.52 − 4.02i)22-s + 6.96·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + (−0.750 − 0.660i)4-s − 0.447i·5-s − 0.457i·7-s + (0.883 − 0.468i)8-s + (0.418 + 0.157i)10-s − 0.917·11-s − 1.40·13-s + (0.427 + 0.161i)14-s + (0.126 + 0.991i)16-s + 0.543i·17-s − 0.582i·19-s + (−0.295 + 0.335i)20-s + (0.324 − 0.858i)22-s + 1.45·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.750 - 0.660i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ -0.750 - 0.660i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6310391043\)
\(L(\frac12)\) \(\approx\) \(0.6310391043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.499 - 1.32i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 1.20iT - 7T^{2} \)
11 \( 1 + 3.04T + 11T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 2.23iT - 17T^{2} \)
19 \( 1 + 2.53iT - 19T^{2} \)
23 \( 1 - 6.96T + 23T^{2} \)
29 \( 1 - 5.59iT - 29T^{2} \)
31 \( 1 - 3.27iT - 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 4.27iT - 41T^{2} \)
43 \( 1 - 4.31iT - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 9.42iT - 53T^{2} \)
59 \( 1 - 8.92T + 59T^{2} \)
61 \( 1 + 2.65T + 61T^{2} \)
67 \( 1 - 14.4iT - 67T^{2} \)
71 \( 1 - 3.49T + 71T^{2} \)
73 \( 1 - 5.10T + 73T^{2} \)
79 \( 1 - 3.76iT - 79T^{2} \)
83 \( 1 - 0.572T + 83T^{2} \)
89 \( 1 - 2.40iT - 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−9.457649525699748658630340841752, −8.856196864190536503316936993298, −8.050807311819937801884128386797, −7.20424812808375450813471893900, −6.83520373023715833126850999936, −5.39555846806467749533591601050, −5.09586902727984126206281911901, −4.13186318088180406788972583615, −2.70408359102789883746457369826, −1.13116941027696715065001763549, 0.30697466361927593168173784398, 2.14848965459851938915311227378, 2.70631346875561889561522298753, 3.74770439703656009985778078044, 4.94345806843453493257476089185, 5.49192970011313311334027987479, 7.00911755885803590826905355446, 7.56207426418325648674104850485, 8.443913087792866898171272258699, 9.274101853907423881314365039607

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