Properties

Label 2-28e2-28.27-c3-0-12
Degree $2$
Conductor $784$
Sign $-0.156 - 0.987i$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.34·3-s + 6.94i·5-s + 13.3·9-s + 2.01i·11-s − 11.3i·13-s − 44.0i·15-s + 71.5i·17-s + 81.8·19-s − 192. i·23-s + 76.7·25-s + 86.9·27-s − 92.2·29-s − 252.·31-s − 12.7i·33-s + 277.·37-s + ⋯
L(s)  = 1  − 1.22·3-s + 0.620i·5-s + 0.492·9-s + 0.0551i·11-s − 0.242i·13-s − 0.758i·15-s + 1.02i·17-s + 0.988·19-s − 1.74i·23-s + 0.614·25-s + 0.619·27-s − 0.590·29-s − 1.46·31-s − 0.0673i·33-s + 1.23·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ -0.156 - 0.987i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8520684054\)
\(L(\frac12)\) \(\approx\) \(0.8520684054\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3 \( 1 + 6.34T + 27T^{2} \)
5 \( 1 - 6.94iT - 125T^{2} \)
11 \( 1 - 2.01iT - 1.33e3T^{2} \)
13 \( 1 + 11.3iT - 2.19e3T^{2} \)
17 \( 1 - 71.5iT - 4.91e3T^{2} \)
19 \( 1 - 81.8T + 6.85e3T^{2} \)
23 \( 1 + 192. iT - 1.21e4T^{2} \)
29 \( 1 + 92.2T + 2.43e4T^{2} \)
31 \( 1 + 252.T + 2.97e4T^{2} \)
37 \( 1 - 277.T + 5.06e4T^{2} \)
41 \( 1 - 276. iT - 6.89e4T^{2} \)
43 \( 1 + 418. iT - 7.95e4T^{2} \)
47 \( 1 - 416.T + 1.03e5T^{2} \)
53 \( 1 + 310.T + 1.48e5T^{2} \)
59 \( 1 + 184.T + 2.05e5T^{2} \)
61 \( 1 + 148. iT - 2.26e5T^{2} \)
67 \( 1 + 180. iT - 3.00e5T^{2} \)
71 \( 1 - 456. iT - 3.57e5T^{2} \)
73 \( 1 - 874. iT - 3.89e5T^{2} \)
79 \( 1 - 428. iT - 4.93e5T^{2} \)
83 \( 1 - 337.T + 5.71e5T^{2} \)
89 \( 1 + 35.0iT - 7.04e5T^{2} \)
97 \( 1 - 1.60e3iT - 9.12e5T^{2} \)
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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

−10.57664185332401646923475675577, −9.443105788056071074167331371017, −8.381884268269670361795935327286, −7.33285196950574081832424315324, −6.50550382387358380905847408619, −5.80436933643085990769454520965, −4.94558838249981157381511219472, −3.78530249510133930228166582496, −2.51479501340883220471885827317, −0.928804277201356343153907793326, 0.36217215043425742557695822885, 1.44783738433954088060115158626, 3.15047351761297301864910652534, 4.48694272511961303165969095111, 5.35302234529895197483006614817, 5.84032305375380235060956642258, 7.06562312262475115768783959881, 7.73371182668291502153100889225, 9.141619492538627517202371648255, 9.510576705923569915041102311812

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