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Properties

Label 2-1248-104.99-c1-0-2
Degree $2$
Conductor $1248$
Sign $-0.385 - 0.922i$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  + 3-s + (0.220 − 0.220i)5-s + (−0.834 − 0.834i)7-s + 9-s + (−4.19 + 4.19i)11-s + (−1.68 + 3.18i)13-s + (0.220 − 0.220i)15-s + 5.40i·17-s + (−1.07 − 1.07i)19-s + (−0.834 − 0.834i)21-s − 8.62·23-s + 4.90i·25-s + 27-s − 7.07i·29-s + (0.834 − 0.834i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.0986 − 0.0986i)5-s + (−0.315 − 0.315i)7-s + 0.333·9-s + (−1.26 + 1.26i)11-s + (−0.466 + 0.884i)13-s + (0.0569 − 0.0569i)15-s + 1.31i·17-s + (−0.247 − 0.247i)19-s + (−0.182 − 0.182i)21-s − 1.79·23-s + 0.980i·25-s + 0.192·27-s − 1.31i·29-s + (0.149 − 0.149i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.385 - 0.922i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.385 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.133396588\)
\(L(\frac12)\) \(\approx\) \(1.133396588\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + (1.68 - 3.18i)T \)
good5 \( 1 + (-0.220 + 0.220i)T - 5iT^{2} \)
7 \( 1 + (0.834 + 0.834i)T + 7iT^{2} \)
11 \( 1 + (4.19 - 4.19i)T - 11iT^{2} \)
17 \( 1 - 5.40iT - 17T^{2} \)
19 \( 1 + (1.07 + 1.07i)T + 19iT^{2} \)
23 \( 1 + 8.62T + 23T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 + (-0.834 + 0.834i)T - 31iT^{2} \)
37 \( 1 + (-3.93 - 3.93i)T + 37iT^{2} \)
41 \( 1 + (-7.68 - 7.68i)T + 41iT^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 + (-6.33 - 6.33i)T + 47iT^{2} \)
53 \( 1 + 0.517iT - 53T^{2} \)
59 \( 1 + (-3.64 + 3.64i)T - 59iT^{2} \)
61 \( 1 + 8.80iT - 61T^{2} \)
67 \( 1 + (3.20 + 3.20i)T + 67iT^{2} \)
71 \( 1 + (9.26 - 9.26i)T - 71iT^{2} \)
73 \( 1 + (1.70 - 1.70i)T - 73iT^{2} \)
79 \( 1 + 10.7iT - 79T^{2} \)
83 \( 1 + (-5.49 - 5.49i)T + 83iT^{2} \)
89 \( 1 + (2.33 - 2.33i)T - 89iT^{2} \)
97 \( 1 + (1.70 + 1.70i)T + 97iT^{2} \)
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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

−9.838821791596161959911449488344, −9.375567534788422550801758582185, −8.036579009099842781726936042985, −7.78687988825608929365432238866, −6.72465086100465236922423487102, −5.85220223754312486939750802892, −4.57191489282535504799385449331, −4.04014888095498675309217287373, −2.61277549733369926249165014896, −1.81820022143677897911585984434, 0.40933823569086579571709198309, 2.44191853936139891887604609704, 2.94326283526520621135762812821, 4.11712811285825008614742596669, 5.41900123781536151224424094993, 5.88708451423160249226166953743, 7.18924680057472920793465913168, 7.889240720362724636341006834721, 8.586393039549980166760316318433, 9.375239726181689857637141067026

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