Properties

Label 2-90e2-5.4-c1-0-36
Degree $2$
Conductor $8100$
Sign $0.447 + 0.894i$
Analytic cond. $64.6788$
Root an. cond. $8.04231$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·7-s − 6·11-s + 5i·13-s + 3i·17-s − 2·19-s + 6i·23-s − 3·29-s − 4·31-s − 5i·37-s − 6·41-s − 10i·43-s + 3·49-s − 6i·53-s + 12·59-s + 5·61-s + ⋯
L(s)  = 1  − 0.755i·7-s − 1.80·11-s + 1.38i·13-s + 0.727i·17-s − 0.458·19-s + 1.25i·23-s − 0.557·29-s − 0.718·31-s − 0.821i·37-s − 0.937·41-s − 1.52i·43-s + 0.428·49-s − 0.824i·53-s + 1.56·59-s + 0.640·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(64.6788\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8100} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8100,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9942595267\)
\(L(\frac12)\) \(\approx\) \(0.9942595267\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 5iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−7.51286835482754762533081633188, −7.21591532347231014218535267455, −6.39289644502238233384097934576, −5.45958955349260255145868380700, −5.04607003206315903452143038662, −3.94209099472952464764194416203, −3.63130951905797775119081344365, −2.31876788625713483086797959618, −1.77257783546612488084048496557, −0.31566750087458468002636643968, 0.70567478024749120046517004194, 2.25546324494438266553196219455, 2.69085462913392214965185512487, 3.43986437011666863051361522215, 4.66208884705078039334772759680, 5.23292382961246028370071152511, 5.69256116737658238253300917196, 6.52249938862100524283843083494, 7.39595517181118366384425852162, 8.043646670752233854528589348122

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