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Properties

Label 2-1248-52.47-c1-0-3
Degree $2$
Conductor $1248$
Sign $0.957 - 0.289i$
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  − i·3-s + (−2.91 − 2.91i)5-s + (3.51 + 3.51i)7-s − 9-s + (0.406 + 0.406i)11-s + (2 + 3i)13-s + (−2.91 + 2.91i)15-s + 2.81i·17-s + (−4.32 + 4.32i)19-s + (3.51 − 3.51i)21-s + 4·23-s + 12.0i·25-s + i·27-s − 3.83·29-s + (4.32 − 4.32i)31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−1.30 − 1.30i)5-s + (1.32 + 1.32i)7-s − 0.333·9-s + (0.122 + 0.122i)11-s + (0.554 + 0.832i)13-s + (−0.753 + 0.753i)15-s + 0.682i·17-s + (−0.991 + 0.991i)19-s + (0.766 − 0.766i)21-s + 0.834·23-s + 2.40i·25-s + 0.192i·27-s − 0.712·29-s + (0.776 − 0.776i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.367005994\)
\(L(\frac12)\) \(\approx\) \(1.367005994\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-2 - 3i)T \)
good5 \( 1 + (2.91 + 2.91i)T + 5iT^{2} \)
7 \( 1 + (-3.51 - 3.51i)T + 7iT^{2} \)
11 \( 1 + (-0.406 - 0.406i)T + 11iT^{2} \)
17 \( 1 - 2.81iT - 17T^{2} \)
19 \( 1 + (4.32 - 4.32i)T - 19iT^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 3.83T + 29T^{2} \)
31 \( 1 + (-4.32 + 4.32i)T - 31iT^{2} \)
37 \( 1 + (-1.81 + 1.81i)T - 37iT^{2} \)
41 \( 1 + (-6.10 - 6.10i)T + 41iT^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (3.42 + 3.42i)T + 47iT^{2} \)
53 \( 1 + 1.18T + 53T^{2} \)
59 \( 1 + (-7.42 - 7.42i)T + 59iT^{2} \)
61 \( 1 - 9.02T + 61T^{2} \)
67 \( 1 + (-0.489 + 0.489i)T - 67iT^{2} \)
71 \( 1 + (-4.40 + 4.40i)T - 71iT^{2} \)
73 \( 1 + (-4.83 + 4.83i)T - 73iT^{2} \)
79 \( 1 + 7.66iT - 79T^{2} \)
83 \( 1 + (-6.61 + 6.61i)T - 83iT^{2} \)
89 \( 1 + (8.91 - 8.91i)T - 89iT^{2} \)
97 \( 1 + (9.85 + 9.85i)T + 97iT^{2} \)
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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

−9.283268915298219919430953161305, −8.689370165524390205651727079063, −8.172527619305224449043834363550, −7.68605965040860452093562387546, −6.33735182465244257937218489715, −5.44437764227710689600982927198, −4.58594758393311878844099207736, −3.86774908116993367854210003813, −2.13965353789861181745406504685, −1.23219661423541604084643899445, 0.68192284649187182172803140415, 2.68935246957197763312485301481, 3.68041980863432133355522531321, 4.30013530294707283287005981602, 5.16323363199478514122035327842, 6.68984066362376324568190264442, 7.21055709625015670229516978979, 8.026762169369924099403417299575, 8.563140285696626287162531192767, 9.958754630289912247852040599876

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