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Properties

Label 2-28e2-7.6-c4-0-16
Degree $2$
Conductor $784$
Sign $-0.156 + 0.987i$
Analytic cond. $81.0420$
Root an. cond. $9.00233$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.9i·3-s + 27.8i·5-s − 174.·9-s + 139.·11-s + 101. i·13-s − 445.·15-s + 542. i·17-s + 139. i·19-s − 229.·23-s − 150.·25-s − 1.50e3i·27-s − 383.·29-s − 397. i·31-s + 2.23e3i·33-s − 898.·37-s + ⋯
L(s)  = 1  + 1.77i·3-s + 1.11i·5-s − 2.15·9-s + 1.15·11-s + 0.603i·13-s − 1.97·15-s + 1.87i·17-s + 0.387i·19-s − 0.434·23-s − 0.240·25-s − 2.05i·27-s − 0.455·29-s − 0.413i·31-s + 2.05i·33-s − 0.656·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(81.0420\)
Root analytic conductor: \(9.00233\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :2),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.476950815\)
\(L(\frac12)\) \(\approx\) \(1.476950815\)
\(L(3)\) 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 \( 1 - 15.9iT - 81T^{2} \)
5 \( 1 - 27.8iT - 625T^{2} \)
11 \( 1 - 139.T + 1.46e4T^{2} \)
13 \( 1 - 101. iT - 2.85e4T^{2} \)
17 \( 1 - 542. iT - 8.35e4T^{2} \)
19 \( 1 - 139. iT - 1.30e5T^{2} \)
23 \( 1 + 229.T + 2.79e5T^{2} \)
29 \( 1 + 383.T + 7.07e5T^{2} \)
31 \( 1 + 397. iT - 9.23e5T^{2} \)
37 \( 1 + 898.T + 1.87e6T^{2} \)
41 \( 1 - 657. iT - 2.82e6T^{2} \)
43 \( 1 + 1.15e3T + 3.41e6T^{2} \)
47 \( 1 - 1.29e3iT - 4.87e6T^{2} \)
53 \( 1 + 5.16e3T + 7.89e6T^{2} \)
59 \( 1 + 4.15e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.84e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.31e3T + 2.01e7T^{2} \)
71 \( 1 + 1.26e3T + 2.54e7T^{2} \)
73 \( 1 + 3.99e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.06e4T + 3.89e7T^{2} \)
83 \( 1 + 5.75e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.18e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.81e3iT - 8.85e7T^{2} \)
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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

−10.30411471861794425667807551987, −9.651941193845224948423622456508, −8.897835223058174961198492339459, −7.975519474924763881222612882178, −6.53820874490246058498946874773, −6.03421213471859166949727349275, −4.73878610927881458468485491147, −3.79504993117872991242934837098, −3.41024702136018831306803819202, −1.90182436802328393057018087007, 0.37414595135672097112321387059, 1.03660491008783978423766119114, 2.00848673354322839256653527170, 3.23877372711743745050830852176, 4.79958069769054424911054352004, 5.63922732860856428851850825392, 6.69088027749462480993967115484, 7.27467321020846696254895285186, 8.210970915102502602799974891487, 8.884178546927118605425103097667

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