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Properties

Label 2-1620-45.14-c2-0-6
Degree $2$
Conductor $1620$
Sign $-0.944 - 0.329i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.29 + 2.55i)5-s + (2.42 − 1.39i)7-s + (−15.7 + 9.07i)11-s + (19.9 + 11.5i)13-s − 5.72·17-s + 23.1·19-s + (0.135 − 0.235i)23-s + (11.8 − 21.9i)25-s + (34.4 − 19.8i)29-s + (−23.6 + 41.0i)31-s + (−6.82 + 12.2i)35-s − 34.8i·37-s + (−11.4 − 6.63i)41-s + (−40.4 + 23.3i)43-s + (−20.4 − 35.4i)47-s + ⋯
L(s)  = 1  + (−0.859 + 0.511i)5-s + (0.345 − 0.199i)7-s + (−1.42 + 0.824i)11-s + (1.53 + 0.885i)13-s − 0.336·17-s + 1.22·19-s + (0.00590 − 0.0102i)23-s + (0.475 − 0.879i)25-s + (1.18 − 0.686i)29-s + (−0.764 + 1.32i)31-s + (−0.194 + 0.348i)35-s − 0.942i·37-s + (−0.280 − 0.161i)41-s + (−0.940 + 0.543i)43-s + (−0.435 − 0.753i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7461051981\)
\(L(\frac12)\) \(\approx\) \(0.7461051981\)
\(L(2)\) 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 \( 1 + (4.29 - 2.55i)T \)
good7 \( 1 + (-2.42 + 1.39i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.7 - 9.07i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-19.9 - 11.5i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 5.72T + 289T^{2} \)
19 \( 1 - 23.1T + 361T^{2} \)
23 \( 1 + (-0.135 + 0.235i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-34.4 + 19.8i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (23.6 - 41.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 34.8iT - 1.36e3T^{2} \)
41 \( 1 + (11.4 + 6.63i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (40.4 - 23.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (20.4 + 35.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 91.3T + 2.80e3T^{2} \)
59 \( 1 + (-68.2 - 39.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.5 + 27.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.98 - 3.45i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 81.5iT - 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + (31.7 + 55.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (0.142 + 0.246i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 28.5iT - 7.92e3T^{2} \)
97 \( 1 + (80.4 - 46.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.627509304875441782284927630826, −8.534851354475816141429913293541, −8.010089997764087187673536923983, −7.17592706264636176995169356680, −6.56824079594060914776295568109, −5.35484161702478033030592781335, −4.53314390139521915492406575367, −3.64963194233573135839246710035, −2.69701529393741040415782136173, −1.39069526861945396801909255515, 0.21960836609003052322858508656, 1.31546278256721780436118786834, 2.98786306064295390816670749962, 3.56115910347085332928728209966, 4.86359602559088627642841157490, 5.40509742945419325510763861914, 6.33281108388014310217383148987, 7.55082551162682358032296882564, 8.254679440991210444190907905694, 8.451666019808530058388545483645

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