Properties

Label 2-1456-7.4-c1-0-18
Degree $2$
Conductor $1456$
Sign $0.608 - 0.793i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−0.673 + 1.16i)3-s + (−1.09 − 1.89i)5-s + (2.19 + 1.47i)7-s + (0.593 + 1.02i)9-s + (−0.524 + 0.907i)11-s + 13-s + 2.94·15-s + (2.64 − 4.58i)17-s + (0.378 + 0.655i)19-s + (−3.19 + 1.56i)21-s + (0.326 + 0.566i)23-s + (0.108 − 0.187i)25-s − 5.63·27-s − 3.10·29-s + (0.513 − 0.890i)31-s + ⋯
L(s)  = 1  + (−0.388 + 0.673i)3-s + (−0.489 − 0.847i)5-s + (0.830 + 0.557i)7-s + (0.197 + 0.342i)9-s + (−0.158 + 0.273i)11-s + 0.277·13-s + 0.760·15-s + (0.641 − 1.11i)17-s + (0.0868 + 0.150i)19-s + (−0.697 + 0.342i)21-s + (0.0681 + 0.118i)23-s + (0.0216 − 0.0374i)25-s − 1.08·27-s − 0.576·29-s + (0.0923 − 0.159i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.608 - 0.793i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.491527882\)
\(L(\frac12)\) \(\approx\) \(1.491527882\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.19 - 1.47i)T \)
13 \( 1 - T \)
good3 \( 1 + (0.673 - 1.16i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.09 + 1.89i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.524 - 0.907i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.64 + 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.655i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.326 - 0.566i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 + (-0.513 + 0.890i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.44 - 9.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.32T + 41T^{2} \)
43 \( 1 + 0.887T + 43T^{2} \)
47 \( 1 + (-1.16 - 2.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.524 - 0.907i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.24 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.23 + 3.87i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.60T + 71T^{2} \)
73 \( 1 + (-4.14 + 7.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.07 - 1.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.66T + 83T^{2} \)
89 \( 1 + (-2.88 - 4.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.88T + 97T^{2} \)
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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

−9.577173771692324857583616881756, −8.924476583723686215172755062542, −7.942009275346408450419453159662, −7.56561968788679978878790525442, −6.12337790776599500508544987347, −5.11705954964623065246067199486, −4.82427709313003046682975584647, −3.91807465701336303548953063270, −2.50434919060864698465100652505, −1.10586284427429639946019474789, 0.797064409424662105347711011215, 1.99744553651236063430381888473, 3.46082512436929873513478112193, 4.10962044137147676843386232638, 5.42682124606049780852683396827, 6.24495822327219092172830144802, 7.07366177486095271401612070667, 7.64887519768983054159889633116, 8.309534612038215372912501823129, 9.447364921865855320361401993771

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