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Properties

Label 2-1620-45.14-c2-0-19
Degree $2$
Conductor $1620$
Sign $0.787 + 0.615i$
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  + (−4.45 + 2.26i)5-s + (−2.97 + 1.71i)7-s + (−11.1 + 6.41i)11-s + (1.55 + 0.896i)13-s − 30.9·17-s − 19.2·19-s + (−1.19 + 2.06i)23-s + (14.6 − 20.2i)25-s + (31.0 − 17.9i)29-s + (20.4 − 35.4i)31-s + (9.36 − 14.4i)35-s + 53.6i·37-s + (−2.14 − 1.24i)41-s + (47.7 − 27.5i)43-s + (28.5 + 49.4i)47-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)5-s + (−0.425 + 0.245i)7-s + (−1.00 + 0.582i)11-s + (0.119 + 0.0689i)13-s − 1.82·17-s − 1.01·19-s + (−0.0518 + 0.0897i)23-s + (0.587 − 0.808i)25-s + (1.07 − 0.618i)29-s + (0.659 − 1.14i)31-s + (0.267 − 0.411i)35-s + 1.44i·37-s + (−0.0524 − 0.0302i)41-s + (1.11 − 0.641i)43-s + (0.606 + 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.787 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.787 + 0.615i$
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: \((2,\ 1620,\ (\ :1),\ 0.787 + 0.615i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6699079518\)
\(L(\frac12)\) \(\approx\) \(0.6699079518\)
\(L(2)\) 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 \( 1 + (4.45 - 2.26i)T \)
good7 \( 1 + (2.97 - 1.71i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (11.1 - 6.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.55 - 0.896i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 30.9T + 289T^{2} \)
19 \( 1 + 19.2T + 361T^{2} \)
23 \( 1 + (1.19 - 2.06i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-31.0 + 17.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-20.4 + 35.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 53.6iT - 1.36e3T^{2} \)
41 \( 1 + (2.14 + 1.24i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-47.7 + 27.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-28.5 - 49.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 19.4T + 2.80e3T^{2} \)
59 \( 1 + (59.8 + 34.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.80 + 15.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-44.3 - 25.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.9iT - 5.04e3T^{2} \)
73 \( 1 + 42.7iT - 5.32e3T^{2} \)
79 \( 1 + (44.4 + 76.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-14.0 - 24.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 68.3iT - 7.92e3T^{2} \)
97 \( 1 + (-135. + 78.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.038919073044478490665726381679, −8.244517194341729778662810685447, −7.61340792293909861704871592174, −6.63806111280255579732115781211, −6.14535459567632057924557737070, −4.65125449730511718724440217493, −4.27716859448306727498412711886, −2.92365185725490341958233146894, −2.24971345743174093916533741522, −0.29013630249823298659181719344, 0.64870201374477334088954749226, 2.31368474658884938761508692528, 3.34924040415027878636004867488, 4.33195413549970036263017042714, 4.97674360807724726297657323367, 6.15350841411146431918994182919, 6.91561781689298905965972215419, 7.76128376059566234096481029010, 8.700880482280668809741295428598, 8.871780865177626469083155751059

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