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Properties

Label 2-1620-12.11-c1-0-20
Degree $2$
Conductor $1620$
Sign $-0.636 - 0.770i$
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  + (0.478 + 1.33i)2-s + (−1.54 + 1.27i)4-s − i·5-s − 2.36i·7-s + (−2.43 − 1.44i)8-s + (1.33 − 0.478i)10-s − 3.15·11-s + 1.48·13-s + (3.15 − 1.13i)14-s + (0.754 − 3.92i)16-s + 3.53i·17-s + 8.30i·19-s + (1.27 + 1.54i)20-s + (−1.50 − 4.19i)22-s + 2.03·23-s + ⋯
L(s)  = 1  + (0.338 + 0.940i)2-s + (−0.770 + 0.636i)4-s − 0.447i·5-s − 0.895i·7-s + (−0.860 − 0.509i)8-s + (0.420 − 0.151i)10-s − 0.951·11-s + 0.411·13-s + (0.842 − 0.303i)14-s + (0.188 − 0.982i)16-s + 0.857i·17-s + 1.90i·19-s + (0.284 + 0.344i)20-s + (−0.321 − 0.895i)22-s + 0.423·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.636 - 0.770i$
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: \((2,\ 1620,\ (\ :1/2),\ -0.636 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.375137124\)
\(L(\frac12)\) \(\approx\) \(1.375137124\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.478 - 1.33i)T \)
3 \( 1 \)
5 \( 1 + iT \)
good7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 + 3.15T + 11T^{2} \)
13 \( 1 - 1.48T + 13T^{2} \)
17 \( 1 - 3.53iT - 17T^{2} \)
19 \( 1 - 8.30iT - 19T^{2} \)
23 \( 1 - 2.03T + 23T^{2} \)
29 \( 1 + 0.315iT - 29T^{2} \)
31 \( 1 - 6.46iT - 31T^{2} \)
37 \( 1 - 7.56T + 37T^{2} \)
41 \( 1 - 6.32iT - 41T^{2} \)
43 \( 1 - 9.58iT - 43T^{2} \)
47 \( 1 - 3.63T + 47T^{2} \)
53 \( 1 - 12.3iT - 53T^{2} \)
59 \( 1 + 6.81T + 59T^{2} \)
61 \( 1 - 5.64T + 61T^{2} \)
67 \( 1 + 7.60iT - 67T^{2} \)
71 \( 1 + 5.95T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 5.96iT - 79T^{2} \)
83 \( 1 + 5.34T + 83T^{2} \)
89 \( 1 - 3.28iT - 89T^{2} \)
97 \( 1 + 2.42T + 97T^{2} \)
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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

−9.602937206908999804411095978315, −8.558702146904444574850679038552, −7.953618677916759145751959898676, −7.44815811019333510110180284732, −6.29779734250499991999712510492, −5.75801029459593459485686823668, −4.71911762391605253492851612063, −4.03156778680448499639593639961, −3.08860732580100346291112294710, −1.24923043640067456172329333655, 0.51989601423744474893026682963, 2.39523288381685441584764747195, 2.66856775890233346347187547867, 3.88596173549073592241475373419, 5.02172091168972701051523218069, 5.51373525397640127495188076223, 6.55859857601535734536877062848, 7.54557680857669527058983524949, 8.654567714235504801125628319058, 9.187621507131896054643844488799

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