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Properties

Label 2-1620-9.5-c2-0-22
Degree $2$
Conductor $1620$
Sign $-0.0871 + 0.996i$
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  + (1.93 + 1.11i)5-s + (−2.26 − 3.92i)7-s + (−12.9 + 7.46i)11-s + (−7.82 + 13.5i)13-s − 7.53i·17-s + 9.77·19-s + (9.38 + 5.41i)23-s + (2.5 + 4.33i)25-s + (−3.94 + 2.27i)29-s + (24.4 − 42.4i)31-s − 10.1i·35-s + 21.4·37-s + (−45.6 − 26.3i)41-s + (−20.5 − 35.6i)43-s + (32.2 − 18.6i)47-s + ⋯
L(s)  = 1  + (0.387 + 0.223i)5-s + (−0.323 − 0.560i)7-s + (−1.17 + 0.678i)11-s + (−0.602 + 1.04i)13-s − 0.443i·17-s + 0.514·19-s + (0.408 + 0.235i)23-s + (0.100 + 0.173i)25-s + (−0.136 + 0.0785i)29-s + (0.789 − 1.36i)31-s − 0.289i·35-s + 0.580·37-s + (−1.11 − 0.642i)41-s + (−0.478 − 0.829i)43-s + (0.685 − 0.395i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.0871 + 0.996i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.0871 + 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.064228964\)
\(L(\frac12)\) \(\approx\) \(1.064228964\)
\(L(2)\) 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 \( 1 + (-1.93 - 1.11i)T \)
good7 \( 1 + (2.26 + 3.92i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.9 - 7.46i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (7.82 - 13.5i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 7.53iT - 289T^{2} \)
19 \( 1 - 9.77T + 361T^{2} \)
23 \( 1 + (-9.38 - 5.41i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (3.94 - 2.27i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-24.4 + 42.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 21.4T + 1.36e3T^{2} \)
41 \( 1 + (45.6 + 26.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (20.5 + 35.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-32.2 + 18.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 81.8iT - 2.80e3T^{2} \)
59 \( 1 + (-70.3 - 40.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.5 + 42.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-23.2 + 40.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 17.4iT - 5.04e3T^{2} \)
73 \( 1 + 101.T + 5.32e3T^{2} \)
79 \( 1 + (28.7 + 49.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (75.2 - 43.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 41.3iT - 7.92e3T^{2} \)
97 \( 1 + (90.4 + 156. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.164324601834435455289373146447, −8.119702019037484861598711767988, −7.17107196862252877114532093030, −6.88780710510858958596037555479, −5.63301301305166057285268201194, −4.89918062695747438303257504183, −3.95791934248057978046571695931, −2.77378907239261017766177196515, −1.92574012774867587436263219205, −0.30504947971517332960104638929, 1.09659428524908615696932748077, 2.66473943233404213921723637181, 3.08628855325437017956816574696, 4.62289819704443476186935764605, 5.44790505588317969208602611166, 5.94963286862750095752742890031, 7.03546648047146137981206529449, 8.007368990127976801182792975329, 8.522705488214662150260141644844, 9.448877351687454659439433452845

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