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Properties

Label 2-28e2-28.27-c3-0-59
Degree $2$
Conductor $784$
Sign $-0.810 - 0.585i$
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  + 3.30·3-s − 13.4i·5-s − 16.0·9-s − 28.0i·11-s + 27.3i·13-s − 44.4i·15-s − 1.11i·17-s − 150.·19-s + 67.4i·23-s − 56.2·25-s − 142.·27-s + 217.·29-s − 42.9·31-s − 92.6i·33-s − 238.·37-s + ⋯
L(s)  = 1  + 0.635·3-s − 1.20i·5-s − 0.595·9-s − 0.768i·11-s + 0.583i·13-s − 0.765i·15-s − 0.0158i·17-s − 1.81·19-s + 0.611i·23-s − 0.449·25-s − 1.01·27-s + 1.39·29-s − 0.249·31-s − 0.488i·33-s − 1.05·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.810 - 0.585i$
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: \((2,\ 784,\ (\ :3/2),\ -0.810 - 0.585i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.09655217403\)
\(L(\frac12)\) \(\approx\) \(0.09655217403\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.30T + 27T^{2} \)
5 \( 1 + 13.4iT - 125T^{2} \)
11 \( 1 + 28.0iT - 1.33e3T^{2} \)
13 \( 1 - 27.3iT - 2.19e3T^{2} \)
17 \( 1 + 1.11iT - 4.91e3T^{2} \)
19 \( 1 + 150.T + 6.85e3T^{2} \)
23 \( 1 - 67.4iT - 1.21e4T^{2} \)
29 \( 1 - 217.T + 2.43e4T^{2} \)
31 \( 1 + 42.9T + 2.97e4T^{2} \)
37 \( 1 + 238.T + 5.06e4T^{2} \)
41 \( 1 - 21.4iT - 6.89e4T^{2} \)
43 \( 1 - 219. iT - 7.95e4T^{2} \)
47 \( 1 + 156.T + 1.03e5T^{2} \)
53 \( 1 + 666.T + 1.48e5T^{2} \)
59 \( 1 - 782.T + 2.05e5T^{2} \)
61 \( 1 - 112. iT - 2.26e5T^{2} \)
67 \( 1 - 1.02e3iT - 3.00e5T^{2} \)
71 \( 1 - 621. iT - 3.57e5T^{2} \)
73 \( 1 - 272. iT - 3.89e5T^{2} \)
79 \( 1 + 837. iT - 4.93e5T^{2} \)
83 \( 1 + 105.T + 5.71e5T^{2} \)
89 \( 1 + 1.00e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.84e3iT - 9.12e5T^{2} \)
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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

−9.054656862239998032260234065787, −8.586315476068403150320752000986, −8.108352919592354205046797764785, −6.72976264430736663791669522973, −5.78268825549117842764383428431, −4.79143209307302767086075171256, −3.85815678629768142299071825651, −2.66361294237535927862092215158, −1.42881388033605235061646281825, −0.02208551845198447728129295290, 2.08596029389494214648195598002, 2.82786725050769216344553982164, 3.79215452635797678582200572985, 5.00767664984680458864266511785, 6.32799782786494612017074124763, 6.85194636644203593237193170844, 7.981013468757299522405051488144, 8.560438823462512371546406078512, 9.605978922114789314177114166053, 10.57468656099766321807869552346

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