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Properties

Label 2-1620-45.29-c2-0-20
Degree $2$
Conductor $1620$
Sign $0.578 - 0.815i$
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  + (−4.91 − 0.897i)5-s + (10.9 + 6.33i)7-s + (16.1 + 9.35i)11-s + (−5.88 + 3.39i)13-s + 11.5·17-s + 24.2·19-s + (−14.6 − 25.3i)23-s + (23.3 + 8.82i)25-s + (−12.9 − 7.47i)29-s + (−6.91 − 11.9i)31-s + (−48.2 − 40.9i)35-s + 38.2i·37-s + (−26.7 + 15.4i)41-s + (11.3 + 6.53i)43-s + (−2.84 + 4.92i)47-s + ⋯
L(s)  = 1  + (−0.983 − 0.179i)5-s + (1.56 + 0.904i)7-s + (1.47 + 0.850i)11-s + (−0.452 + 0.261i)13-s + 0.679·17-s + 1.27·19-s + (−0.635 − 1.10i)23-s + (0.935 + 0.353i)25-s + (−0.446 − 0.257i)29-s + (−0.223 − 0.386i)31-s + (−1.37 − 1.17i)35-s + 1.03i·37-s + (−0.653 + 0.377i)41-s + (0.263 + 0.152i)43-s + (−0.0605 + 0.104i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.237906730\)
\(L(\frac12)\) \(\approx\) \(2.237906730\)
\(L(2)\) 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 \( 1 + (4.91 + 0.897i)T \)
good7 \( 1 + (-10.9 - 6.33i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-16.1 - 9.35i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.88 - 3.39i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 11.5T + 289T^{2} \)
19 \( 1 - 24.2T + 361T^{2} \)
23 \( 1 + (14.6 + 25.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (12.9 + 7.47i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (6.91 + 11.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 38.2iT - 1.36e3T^{2} \)
41 \( 1 + (26.7 - 15.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.3 - 6.53i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (2.84 - 4.92i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 57.7T + 2.80e3T^{2} \)
59 \( 1 + (-26.2 + 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (38.5 - 66.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-111. + 64.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 35.8iT - 5.04e3T^{2} \)
73 \( 1 + 40.6iT - 5.32e3T^{2} \)
79 \( 1 + (70.0 - 121. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-59.4 + 102. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 75.0iT - 7.92e3T^{2} \)
97 \( 1 + (73.1 + 42.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.229529255416252029642749368503, −8.442822444420836346155302074925, −7.81117270504312355329449772662, −7.14056571890867384413533416173, −6.05911564675841978979490348958, −4.90982523546932884969791147547, −4.53301692043528431114355963193, −3.45772966968659467085981750428, −2.09041256640657043986531758874, −1.14411122428938109249324495470, 0.74570835182690409462262471357, 1.60708159053503352474537484326, 3.42304048752458222295745110108, 3.86049360924706519866220716953, 4.87816450830012150467516215289, 5.68829674634995582042909152540, 7.01518817575452133774538183328, 7.53268220254945280562141157978, 8.122242865225649610827474828757, 8.952741556207050192945776065577

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