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Properties

Label 2-1792-16.13-c1-0-40
Degree $2$
Conductor $1792$
Sign $-0.991 - 0.130i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
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  + (−0.328 − 0.328i)3-s + (−1.40 + 1.40i)5-s − i·7-s − 2.78i·9-s + (−0.444 + 0.444i)11-s + (−2.25 − 2.25i)13-s + 0.919·15-s + 4.85·17-s + (0.114 + 0.114i)19-s + (−0.328 + 0.328i)21-s + 3.20i·23-s + 1.06i·25-s + (−1.89 + 1.89i)27-s + (0.997 + 0.997i)29-s − 5.34·31-s + ⋯
L(s)  = 1  + (−0.189 − 0.189i)3-s + (−0.626 + 0.626i)5-s − 0.377i·7-s − 0.928i·9-s + (−0.133 + 0.133i)11-s + (−0.625 − 0.625i)13-s + 0.237·15-s + 1.17·17-s + (0.0261 + 0.0261i)19-s + (−0.0715 + 0.0715i)21-s + 0.668i·23-s + 0.213i·25-s + (−0.365 + 0.365i)27-s + (0.185 + 0.185i)29-s − 0.959·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.06823327470\)
\(L(\frac12)\) \(\approx\) \(0.06823327470\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 + (0.328 + 0.328i)T + 3iT^{2} \)
5 \( 1 + (1.40 - 1.40i)T - 5iT^{2} \)
11 \( 1 + (0.444 - 0.444i)T - 11iT^{2} \)
13 \( 1 + (2.25 + 2.25i)T + 13iT^{2} \)
17 \( 1 - 4.85T + 17T^{2} \)
19 \( 1 + (-0.114 - 0.114i)T + 19iT^{2} \)
23 \( 1 - 3.20iT - 23T^{2} \)
29 \( 1 + (-0.997 - 0.997i)T + 29iT^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 + (2.03 - 2.03i)T - 37iT^{2} \)
41 \( 1 + 9.57iT - 41T^{2} \)
43 \( 1 + (6.86 - 6.86i)T - 43iT^{2} \)
47 \( 1 + 9.70T + 47T^{2} \)
53 \( 1 + (7.64 - 7.64i)T - 53iT^{2} \)
59 \( 1 + (1.50 - 1.50i)T - 59iT^{2} \)
61 \( 1 + (-1.74 - 1.74i)T + 61iT^{2} \)
67 \( 1 + (8.96 + 8.96i)T + 67iT^{2} \)
71 \( 1 + 7.18iT - 71T^{2} \)
73 \( 1 + 9.04iT - 73T^{2} \)
79 \( 1 + 9.58T + 79T^{2} \)
83 \( 1 + (5.30 + 5.30i)T + 83iT^{2} \)
89 \( 1 - 2.49iT - 89T^{2} \)
97 \( 1 - 5.89T + 97T^{2} \)
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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

−8.916021840185136058016698651474, −7.67559298909664268739799517529, −7.50109858034834384136502526003, −6.56848971792303948096564816426, −5.69637310254353535167033081294, −4.77630336304281912733003851473, −3.48766533754416428539297250488, −3.19131641209381131830169518769, −1.48058813756694946884888447314, −0.02622507490562297477307873735, 1.66376964208160175781251552295, 2.85115824314597269904519172205, 4.02188444450164919367136819931, 4.88952895000227328107802569244, 5.39073266650766868063886315145, 6.50947284306868845709281364420, 7.49585079613164579331264688607, 8.162991700102944933195426140413, 8.743533594788682317295133852259, 9.807803067337186499762822633628

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