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Properties

Label 2-1248-104.83-c1-0-13
Degree $2$
Conductor $1248$
Sign $0.856 - 0.516i$
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 + (1.61 + 1.61i)5-s + (2.08 − 2.08i)7-s + 9-s + (0.0673 + 0.0673i)11-s + (1.04 + 3.45i)13-s + (1.61 + 1.61i)15-s + 7.18i·17-s + (−2.30 + 2.30i)19-s + (2.08 − 2.08i)21-s + 2.00·23-s + 0.246i·25-s + 27-s − 4.37i·29-s + (−2.08 − 2.08i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.724 + 0.724i)5-s + (0.786 − 0.786i)7-s + 0.333·9-s + (0.0203 + 0.0203i)11-s + (0.288 + 0.957i)13-s + (0.418 + 0.418i)15-s + 1.74i·17-s + (−0.528 + 0.528i)19-s + (0.454 − 0.454i)21-s + 0.417·23-s + 0.0493i·25-s + 0.192·27-s − 0.811i·29-s + (−0.373 − 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ 0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.541682230\)
\(L(\frac12)\) \(\approx\) \(2.541682230\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 3.45i)T \)
good5 \( 1 + (-1.61 - 1.61i)T + 5iT^{2} \)
7 \( 1 + (-2.08 + 2.08i)T - 7iT^{2} \)
11 \( 1 + (-0.0673 - 0.0673i)T + 11iT^{2} \)
17 \( 1 - 7.18iT - 17T^{2} \)
19 \( 1 + (2.30 - 2.30i)T - 19iT^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + 4.37iT - 29T^{2} \)
31 \( 1 + (2.08 + 2.08i)T + 31iT^{2} \)
37 \( 1 + (-6.43 + 6.43i)T - 37iT^{2} \)
41 \( 1 + (7.94 - 7.94i)T - 41iT^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 + (-5.31 + 5.31i)T - 47iT^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 + (-3.96 - 3.96i)T + 59iT^{2} \)
61 \( 1 - 1.94iT - 61T^{2} \)
67 \( 1 + (8.16 - 8.16i)T - 67iT^{2} \)
71 \( 1 + (-0.00829 - 0.00829i)T + 71iT^{2} \)
73 \( 1 + (-0.419 - 0.419i)T + 73iT^{2} \)
79 \( 1 - 8.46iT - 79T^{2} \)
83 \( 1 + (-3.35 + 3.35i)T - 83iT^{2} \)
89 \( 1 + (-1.90 - 1.90i)T + 89iT^{2} \)
97 \( 1 + (-8.76 + 8.76i)T - 97iT^{2} \)
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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

−9.928835765930081263793213485983, −8.863625306464134547531919366515, −8.196701031939746818348774525609, −7.32588776185105738932672702968, −6.50899536409148448128246680694, −5.74678933515197219180268222131, −4.31194689787270662450009700968, −3.79787338516535770259154610606, −2.32835166048032906627317176386, −1.56356218592166526544000284559, 1.15029074633504816586867164231, 2.32894675490749011339969759271, 3.22065255868507575676553836382, 4.88233207511604081335356095885, 5.09989814505996423114846100886, 6.20631733120671070368496243737, 7.33997723754756898076770584434, 8.161839618542784844138799366303, 8.990643613291539908012627560119, 9.274642695389650140601203579272

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