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Properties

Label 2-1248-52.31-c1-0-13
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: $\chi_{1248} (31, \cdot )$
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

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

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