Properties

Label 2-6048-24.11-c1-0-30
Degree $2$
Conductor $6048$
Sign $-0.0574 - 0.998i$
Analytic cond. $48.2935$
Root an. cond. $6.94935$
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  + 1.27·5-s + i·7-s + 2.84i·11-s − 0.223i·13-s − 0.397i·17-s + 4.64·19-s + 7.36·23-s − 3.36·25-s − 10.6·29-s + 7.67i·31-s + 1.27i·35-s + 4.74i·37-s + 1.97i·41-s − 7.62·43-s + 11.0·47-s + ⋯
L(s)  = 1  + 0.571·5-s + 0.377i·7-s + 0.858i·11-s − 0.0620i·13-s − 0.0964i·17-s + 1.06·19-s + 1.53·23-s − 0.673·25-s − 1.97·29-s + 1.37i·31-s + 0.216i·35-s + 0.780i·37-s + 0.308i·41-s − 1.16·43-s + 1.61·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0574 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.0574 - 0.998i$
Analytic conductor: \(48.2935\)
Root analytic conductor: \(6.94935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6048,\ (\ :1/2),\ -0.0574 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.869450767\)
\(L(\frac12)\) \(\approx\) \(1.869450767\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 1.27T + 5T^{2} \)
11 \( 1 - 2.84iT - 11T^{2} \)
13 \( 1 + 0.223iT - 13T^{2} \)
17 \( 1 + 0.397iT - 17T^{2} \)
19 \( 1 - 4.64T + 19T^{2} \)
23 \( 1 - 7.36T + 23T^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
31 \( 1 - 7.67iT - 31T^{2} \)
37 \( 1 - 4.74iT - 37T^{2} \)
41 \( 1 - 1.97iT - 41T^{2} \)
43 \( 1 + 7.62T + 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 - 0.295T + 53T^{2} \)
59 \( 1 - 7.25iT - 59T^{2} \)
61 \( 1 + 9.45iT - 61T^{2} \)
67 \( 1 - 3.52T + 67T^{2} \)
71 \( 1 + 8.37T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 5.22iT - 79T^{2} \)
83 \( 1 + 9.11iT - 83T^{2} \)
89 \( 1 - 8.94iT - 89T^{2} \)
97 \( 1 - 0.228T + 97T^{2} \)
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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

−8.269247158136097520552189807818, −7.37786181425472547375048702740, −6.98882288033987509579984899152, −6.07132114695075466562728901553, −5.28000625568993648361005281980, −4.91980182728593635587036795060, −3.75886029911626036801294566968, −2.97662162675423155493943424882, −2.05196193220526394803705989888, −1.22182559670471593948504479392, 0.48294038218327591776266357351, 1.55571167130939036268456911600, 2.54409491628425415808495478554, 3.47999078104056876551494720272, 4.09185521496575901725560033395, 5.30316984254090898558194428936, 5.60927378147316654941612123760, 6.42061167586427800645270898163, 7.33545811464936561462257380801, 7.69022861213046416760166574841

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