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Properties

Label 2-1620-9.7-c3-0-5
Degree $2$
Conductor $1620$
Sign $-0.766 - 0.642i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 + 4.33i)5-s + (13.4 − 23.2i)7-s + (−23.3 + 40.3i)11-s + (−0.605 − 1.04i)13-s + 87.2·17-s − 125.·19-s + (3.58 + 6.20i)23-s + (−12.5 + 21.6i)25-s + (−21.9 + 37.9i)29-s + (−70.1 − 121. i)31-s + 134.·35-s − 187.·37-s + (119. + 207. i)41-s + (−228. + 396. i)43-s + (12.8 − 22.1i)47-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.724 − 1.25i)7-s + (−0.639 + 1.10i)11-s + (−0.0129 − 0.0223i)13-s + 1.24·17-s − 1.51·19-s + (0.0324 + 0.0562i)23-s + (−0.100 + 0.173i)25-s + (−0.140 + 0.243i)29-s + (−0.406 − 0.704i)31-s + 0.647·35-s − 0.831·37-s + (0.456 + 0.791i)41-s + (−0.811 + 1.40i)43-s + (0.0397 − 0.0688i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7486280801\)
\(L(\frac12)\) \(\approx\) \(0.7486280801\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 - 4.33i)T \)
good7 \( 1 + (-13.4 + 23.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (23.3 - 40.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (0.605 + 1.04i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 87.2T + 4.91e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
23 \( 1 + (-3.58 - 6.20i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (21.9 - 37.9i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (70.1 + 121. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 187.T + 5.06e4T^{2} \)
41 \( 1 + (-119. - 207. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (228. - 396. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-12.8 + 22.1i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 82.8T + 1.48e5T^{2} \)
59 \( 1 + (369. + 640. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-14.3 + 24.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-63.1 - 109. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 611.T + 3.57e5T^{2} \)
73 \( 1 - 983.T + 3.89e5T^{2} \)
79 \( 1 + (186. - 322. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (346. - 600. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 873.T + 7.04e5T^{2} \)
97 \( 1 + (-41.3 + 71.6i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.610804713081788621956680105029, −8.291174557431363841544303052407, −7.71337387191134168321929151106, −7.08099903785924126417587316866, −6.20679613634451222263831146323, −5.07124486551506194120822471173, −4.41164904408675548923730292285, −3.46173823967253549778099546277, −2.21406537900170011577736987205, −1.28446791093291406344876863147, 0.15475499907195436424043034037, 1.56793379358235794859788947560, 2.48301331010810289546389536707, 3.51386121595884232228468648708, 4.75412310469415401242385353491, 5.58365955488899775323288827893, 5.92146816371253797698053043779, 7.19231167547340788403424167896, 8.334544753678840952450333177966, 8.496016171403608254403523329851

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