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Properties

Label 2-1620-9.7-c3-0-9
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 + (−11.3 + 19.5i)7-s + (5.48 − 9.50i)11-s + (37.9 + 65.6i)13-s − 15.9·17-s + 58.8·19-s + (−28.3 − 49.0i)23-s + (−12.5 + 21.6i)25-s + (−2.80 + 4.86i)29-s + (60.3 + 104. i)31-s + 113.·35-s + 236.·37-s + (−98.0 − 169. i)41-s + (−9.09 + 15.7i)43-s + (−155. + 270. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.610 + 1.05i)7-s + (0.150 − 0.260i)11-s + (0.808 + 1.40i)13-s − 0.227·17-s + 0.710·19-s + (−0.256 − 0.444i)23-s + (−0.100 + 0.173i)25-s + (−0.0179 + 0.0311i)29-s + (0.349 + 0.605i)31-s + 0.546·35-s + 1.04·37-s + (−0.373 − 0.647i)41-s + (−0.0322 + 0.0558i)43-s + (−0.483 + 0.838i)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

\(L(2)\) \(\approx\) \(1.071309694\)
\(L(\frac12)\) \(\approx\) \(1.071309694\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.3 - 19.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-5.48 + 9.50i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-37.9 - 65.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 15.9T + 4.91e3T^{2} \)
19 \( 1 - 58.8T + 6.85e3T^{2} \)
23 \( 1 + (28.3 + 49.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (2.80 - 4.86i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-60.3 - 104. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 236.T + 5.06e4T^{2} \)
41 \( 1 + (98.0 + 169. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (9.09 - 15.7i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (155. - 270. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 33.3T + 1.48e5T^{2} \)
59 \( 1 + (260. + 451. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-75.1 + 130. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (253. + 438. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 961.T + 3.57e5T^{2} \)
73 \( 1 + 251.T + 3.89e5T^{2} \)
79 \( 1 + (417. - 722. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (345. - 598. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + (642. - 1.11e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.207636112411087743972495632279, −8.715721947607096883102270565926, −7.904879381912537539958506740624, −6.70977277628900024237777444623, −6.21577575414845140683503603751, −5.28058387724147035095702971681, −4.31663946479509068026272547726, −3.39858832375687694422553744538, −2.34326440294617030359863329646, −1.20316334822528967394845319659, 0.25984707521403966053905458945, 1.24769914146808889867007860958, 2.82908572314313599213347925021, 3.56206085598588991731149537214, 4.35368256992059494133438665970, 5.56856064245986030874901804803, 6.33264718740796266837256038499, 7.21665043314003426501084571143, 7.78754962661797053564475840894, 8.626551080591181642046947264208

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