Properties

Label 2-1350-5.4-c3-0-28
Degree $2$
Conductor $1350$
Sign $0.447 + 0.894i$
Analytic cond. $79.6525$
Root an. cond. $8.92482$
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  − 2i·2-s − 4·4-s − 29i·7-s + 8i·8-s − 57·11-s + 20i·13-s − 58·14-s + 16·16-s + 72i·17-s + 106·19-s + 114i·22-s + 174i·23-s + 40·26-s + 116i·28-s + 210·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 1.56i·7-s + 0.353i·8-s − 1.56·11-s + 0.426i·13-s − 1.10·14-s + 0.250·16-s + 1.02i·17-s + 1.27·19-s + 1.10i·22-s + 1.57i·23-s + 0.301·26-s + 0.782i·28-s + 1.34·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(79.6525\)
Root analytic conductor: \(8.92482\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.620226819\)
\(L(\frac12)\) \(\approx\) \(1.620226819\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 29iT - 343T^{2} \)
11 \( 1 + 57T + 1.33e3T^{2} \)
13 \( 1 - 20iT - 2.19e3T^{2} \)
17 \( 1 - 72iT - 4.91e3T^{2} \)
19 \( 1 - 106T + 6.85e3T^{2} \)
23 \( 1 - 174iT - 1.21e4T^{2} \)
29 \( 1 - 210T + 2.43e4T^{2} \)
31 \( 1 - 47T + 2.97e4T^{2} \)
37 \( 1 + 2iT - 5.06e4T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 - 218iT - 7.95e4T^{2} \)
47 \( 1 + 474iT - 1.03e5T^{2} \)
53 \( 1 - 81iT - 1.48e5T^{2} \)
59 \( 1 + 84T + 2.05e5T^{2} \)
61 \( 1 - 56T + 2.26e5T^{2} \)
67 \( 1 - 142iT - 3.00e5T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 + 1.15e3iT - 3.89e5T^{2} \)
79 \( 1 - 160T + 4.93e5T^{2} \)
83 \( 1 - 735iT - 5.71e5T^{2} \)
89 \( 1 - 954T + 7.04e5T^{2} \)
97 \( 1 + 191iT - 9.12e5T^{2} \)
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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

−9.360476442195102426909377625070, −8.112005710953596558754537046565, −7.68446728093418935111293908007, −6.77064568934388662209729383080, −5.52818148011570479624502509953, −4.72152331123398583930310862577, −3.77535007477901229118472222940, −3.00328441577793788160038904869, −1.65801857273428066666728279876, −0.65990025893975089199837236689, 0.61479866302321414715485372095, 2.50199115792477931641273723951, 2.98386905272880015320848510951, 4.73930524845234838634584571955, 5.25525762468887170464745488654, 5.95021969662618112462855131914, 6.92356320768044654902762132567, 7.907735271204015252541322437778, 8.399949848473791044922229643162, 9.236573249697945980053961757728

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