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Properties

Label 2-1248-13.3-c1-0-15
Degree $2$
Conductor $1248$
Sign $-0.0128 + 0.999i$
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  + (−0.5 − 0.866i)3-s − 3.82·5-s + (−1.41 + 2.44i)7-s + (−0.499 + 0.866i)9-s + (1.41 + 2.44i)11-s + (3.5 − 0.866i)13-s + (1.91 + 3.31i)15-s + (−2.91 + 5.04i)17-s + (3.41 − 5.91i)19-s + 2.82·21-s + (−3.41 − 5.91i)23-s + 9.65·25-s + 0.999·27-s + (1.91 + 3.31i)29-s − 9.65·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 1.71·5-s + (−0.534 + 0.925i)7-s + (−0.166 + 0.288i)9-s + (0.426 + 0.738i)11-s + (0.970 − 0.240i)13-s + (0.494 + 0.856i)15-s + (−0.706 + 1.22i)17-s + (0.783 − 1.35i)19-s + 0.617·21-s + (−0.711 − 1.23i)23-s + 1.93·25-s + 0.192·27-s + (0.355 + 0.615i)29-s − 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.0128 + 0.999i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5926949289\)
\(L(\frac12)\) \(\approx\) \(0.5926949289\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.5 + 0.866i)T \)
good5 \( 1 + 3.82T + 5T^{2} \)
7 \( 1 + (1.41 - 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.91 - 5.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.41 + 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.41 + 5.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.91 - 3.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + (4.32 + 7.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.914 + 1.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.41 + 9.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.17T + 47T^{2} \)
53 \( 1 - 1.82T + 53T^{2} \)
59 \( 1 + (-1.17 + 2.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.32 - 4.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.24 + 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.24 + 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.65T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.65 - 4.60i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.069125200736470046428234251867, −8.769920256049043158527759045578, −7.80771803329757238262031230908, −7.02673341877336689790361661551, −6.35712889009891917075667111615, −5.27397499168114576207348072687, −4.16607645408182296330180197283, −3.42837811619914126840458832130, −2.09703253249814295918043911907, −0.33921593940667450902898889801, 0.985446964012410656360537575366, 3.43765456288364028271866006713, 3.64261683831574276193055063615, 4.50197957930930055043699671433, 5.71186252846950146092960673451, 6.71544001660177429882310631532, 7.50031350902209840024033591894, 8.185258411264936838984041844191, 9.114392057624419755283097239659, 9.931653033136251506233617736629

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