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Properties

Label 2-1248-104.101-c1-0-3
Degree $2$
Conductor $1248$
Sign $-0.799 - 0.600i$
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  + (−0.866 + 0.5i)3-s + 2.23·5-s + (−3.43 − 1.98i)7-s + (0.499 − 0.866i)9-s + (−0.252 − 0.436i)11-s + (−2.59 − 2.5i)13-s + (−1.93 + 1.11i)15-s + (−2.93 + 5.08i)17-s + (0.252 − 0.436i)19-s + 3.96·21-s + (4.43 + 7.68i)23-s + 0.999i·27-s + (−8.55 + 4.93i)29-s + 4.47i·31-s + (0.436 + 0.252i)33-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + 0.999·5-s + (−1.29 − 0.749i)7-s + (0.166 − 0.288i)9-s + (−0.0759 − 0.131i)11-s + (−0.720 − 0.693i)13-s + (−0.500 + 0.288i)15-s + (−0.712 + 1.23i)17-s + (0.0578 − 0.100i)19-s + 0.865·21-s + (0.925 + 1.60i)23-s + 0.192i·27-s + (−1.58 + 0.916i)29-s + 0.803i·31-s + (0.0759 + 0.0438i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.799 - 0.600i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.799 - 0.600i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4317023040\)
\(L(\frac12)\) \(\approx\) \(0.4317023040\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + (3.43 + 1.98i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.252 + 0.436i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.252 + 0.436i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.43 - 7.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.55 - 4.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.47iT - 31T^{2} \)
37 \( 1 + (-3.60 - 6.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.06 + 0.614i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.68 + 4.43i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.96iT - 47T^{2} \)
53 \( 1 + 7.87iT - 53T^{2} \)
59 \( 1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.976 + 1.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.30 - 2.48i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 2.96T + 83T^{2} \)
89 \( 1 + (1.74 - 1.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (15 + 8.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.804942415755476974001372223977, −9.677853542753487209267644758590, −8.564316328803382131955625525598, −7.27167920315067793299640417753, −6.69371216752314106753361747489, −5.78478623873322907002185414105, −5.18360457647399166558150217970, −3.85653172864468330631580014359, −3.05007538994652375345312046521, −1.53993924587979476344884377189, 0.18239150485829562950863165791, 2.14425928133228512474777427670, 2.74047863521054067379539658868, 4.33445928343365356281356806243, 5.34284872112368530783519293628, 6.10976626157498070854227680736, 6.69748285271051738102838831015, 7.48778647612961986055395339900, 8.911800546088200342545841555583, 9.481379268056355377909047198624

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