Properties

Label 2-28e2-28.27-c3-0-29
Degree $2$
Conductor $784$
Sign $0.944 + 0.327i$
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  − 9.29·3-s + 19.8i·5-s + 59.3·9-s − 11.8i·11-s + 19.9i·13-s − 184. i·15-s + 1.81i·17-s + 7.32·19-s − 106. i·23-s − 269.·25-s − 300.·27-s − 191.·29-s − 125.·31-s + 110. i·33-s + 316.·37-s + ⋯
L(s)  = 1  − 1.78·3-s + 1.77i·5-s + 2.19·9-s − 0.324i·11-s + 0.426i·13-s − 3.17i·15-s + 0.0259i·17-s + 0.0884·19-s − 0.965i·23-s − 2.15·25-s − 2.14·27-s − 1.22·29-s − 0.725·31-s + 0.580i·33-s + 1.40·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.944 + 0.327i$
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: \((2,\ 784,\ (\ :3/2),\ 0.944 + 0.327i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5957240993\)
\(L(\frac12)\) \(\approx\) \(0.5957240993\)
\(L(\frac{5}{2})\) 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 \( 1 + 9.29T + 27T^{2} \)
5 \( 1 - 19.8iT - 125T^{2} \)
11 \( 1 + 11.8iT - 1.33e3T^{2} \)
13 \( 1 - 19.9iT - 2.19e3T^{2} \)
17 \( 1 - 1.81iT - 4.91e3T^{2} \)
19 \( 1 - 7.32T + 6.85e3T^{2} \)
23 \( 1 + 106. iT - 1.21e4T^{2} \)
29 \( 1 + 191.T + 2.43e4T^{2} \)
31 \( 1 + 125.T + 2.97e4T^{2} \)
37 \( 1 - 316.T + 5.06e4T^{2} \)
41 \( 1 + 321. iT - 6.89e4T^{2} \)
43 \( 1 - 74.3iT - 7.95e4T^{2} \)
47 \( 1 + 154.T + 1.03e5T^{2} \)
53 \( 1 + 319.T + 1.48e5T^{2} \)
59 \( 1 - 61.1T + 2.05e5T^{2} \)
61 \( 1 - 309. iT - 2.26e5T^{2} \)
67 \( 1 + 594. iT - 3.00e5T^{2} \)
71 \( 1 - 48.6iT - 3.57e5T^{2} \)
73 \( 1 + 770. iT - 3.89e5T^{2} \)
79 \( 1 + 960. iT - 4.93e5T^{2} \)
83 \( 1 - 1.06e3T + 5.71e5T^{2} \)
89 \( 1 - 655. iT - 7.04e5T^{2} \)
97 \( 1 + 704. iT - 9.12e5T^{2} \)
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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

−10.25338678975102940256696842619, −9.358719591788273411966771979753, −7.69481640713237675011265759377, −6.98831960195417005767591052875, −6.28826505052562766408741012429, −5.74043957284391773310381567144, −4.52565927415771457769916380108, −3.43547386268966083513596547427, −2.01287703836785589823508837111, −0.31101297883909133383863571838, 0.75237114078050223996833044061, 1.61772421812861114649490361161, 3.96596336055300683599093246538, 4.84647053040762145181068244866, 5.42209288809161638717431596672, 6.08399550060941142499733252867, 7.30352296474761291457548658197, 8.153071909648219836657289436675, 9.413633940576317831175564563278, 9.824732375337781525610645915340

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