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Properties

Label 2-28e2-28.27-c5-0-74
Degree $2$
Conductor $784$
Sign $-0.912 + 0.409i$
Analytic cond. $125.740$
Root an. cond. $11.2134$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16.1·3-s − 48.6i·5-s + 17.7·9-s + 400. i·11-s − 228. i·13-s + 784. i·15-s − 519. i·17-s − 946.·19-s + 1.24e3i·23-s + 762.·25-s + 3.63e3·27-s + 3.63e3·29-s − 2.11e3·31-s − 6.46e3i·33-s − 8.59e3·37-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.869i·5-s + 0.0728·9-s + 0.996i·11-s − 0.375i·13-s + 0.900i·15-s − 0.435i·17-s − 0.601·19-s + 0.488i·23-s + 0.244·25-s + 0.960·27-s + 0.803·29-s − 0.395·31-s − 1.03i·33-s − 1.03·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(125.740\)
Root analytic conductor: \(11.2134\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ -0.912 + 0.409i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5082668855\)
\(L(\frac12)\) \(\approx\) \(0.5082668855\)
\(L(\frac{7}{2})\) 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 \( 1 + 16.1T + 243T^{2} \)
5 \( 1 + 48.6iT - 3.12e3T^{2} \)
11 \( 1 - 400. iT - 1.61e5T^{2} \)
13 \( 1 + 228. iT - 3.71e5T^{2} \)
17 \( 1 + 519. iT - 1.41e6T^{2} \)
19 \( 1 + 946.T + 2.47e6T^{2} \)
23 \( 1 - 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.63e3T + 2.05e7T^{2} \)
31 \( 1 + 2.11e3T + 2.86e7T^{2} \)
37 \( 1 + 8.59e3T + 6.93e7T^{2} \)
41 \( 1 + 1.44e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.45e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.57e4T + 2.29e8T^{2} \)
53 \( 1 - 1.33e4T + 4.18e8T^{2} \)
59 \( 1 - 3.11e4T + 7.14e8T^{2} \)
61 \( 1 + 3.48e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.18e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.93e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.23e3iT - 2.07e9T^{2} \)
79 \( 1 - 5.53e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.04e5T + 3.93e9T^{2} \)
89 \( 1 - 1.30e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.26e4iT - 8.58e9T^{2} \)
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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

−9.141437071233743044075139424393, −8.430136361752281260128603314510, −7.29763797632316330953991255634, −6.49709914618342759408642713871, −5.35592468513602302127858024606, −4.99873797884757969834163454515, −3.90207798298958077018973274335, −2.36974102086084997016597364878, −1.09164443069261593641602547593, −0.16010894023301587000055928775, 0.943248497210041448753122727343, 2.45481477984419678024736118164, 3.47626438933848116742325025274, 4.65054742250502196184025539109, 5.71331734619324997170073778564, 6.36485171345559766119097068354, 7.00603459269414355428284681022, 8.253038603074986541760959666547, 8.989633182265280480893501258744, 10.37183785823910953497281155705

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