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Properties

Label 2-1848-21.20-c1-0-22
Degree $2$
Conductor $1848$
Sign $0.617 - 0.786i$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
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  + (−0.468 − 1.66i)3-s + 2.13·5-s + (1.56 + 2.13i)7-s + (−2.56 + 1.56i)9-s + i·11-s + 2.13i·13-s + (−0.999 − 3.56i)15-s − 2.39·17-s + 4.53i·19-s + (2.83 − 3.60i)21-s + 3.12i·23-s − 0.438·25-s + (3.80 + 3.54i)27-s + 0.561i·29-s + 10.4i·31-s + ⋯
L(s)  = 1  + (−0.270 − 0.962i)3-s + 0.955·5-s + (0.590 + 0.807i)7-s + (−0.853 + 0.520i)9-s + 0.301i·11-s + 0.592i·13-s + (−0.258 − 0.919i)15-s − 0.581·17-s + 1.04i·19-s + (0.617 − 0.786i)21-s + 0.651i·23-s − 0.0876·25-s + (0.731 + 0.681i)27-s + 0.104i·29-s + 1.87i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586761836\)
\(L(\frac12)\) \(\approx\) \(1.586761836\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.468 + 1.66i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
11 \( 1 - iT \)
good5 \( 1 - 2.13T + 5T^{2} \)
13 \( 1 - 2.13iT - 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 - 4.53iT - 19T^{2} \)
23 \( 1 - 3.12iT - 23T^{2} \)
29 \( 1 - 0.561iT - 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 + 4.79T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 7.86T + 47T^{2} \)
53 \( 1 + 8.24iT - 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 3.33iT - 61T^{2} \)
67 \( 1 - 5.43T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 + 1.87iT - 97T^{2} \)
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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

−9.296770985139752003973367855453, −8.473450885546284403584462987885, −7.86482116601806064114301373607, −6.78695301353951667617144305996, −6.25082860305417839002347750621, −5.42673258556957580105454428336, −4.78337518729025434698410131743, −3.19945714787940918530212117336, −1.95971142099124675885075142550, −1.62378799277102046043939950930, 0.58036619419417934050846811608, 2.17096048074125538173144800783, 3.25765407117076116122555968979, 4.39644886018899888777544048633, 4.90418822500114388172697787349, 5.91654749783526744726810410463, 6.47750907428579489940049694316, 7.66993601502573649217172647017, 8.461370435637752607252145154391, 9.362465784865653484243551990016

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