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Properties

Label 2-28e2-7.6-c4-0-44
Degree $2$
Conductor $784$
Sign $0.912 - 0.409i$
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  + 16.1i·3-s + 7.83i·5-s − 179.·9-s − 30.5·11-s − 162. i·13-s − 126.·15-s − 345. i·17-s + 263. i·19-s − 213.·23-s + 563.·25-s − 1.58e3i·27-s − 1.01e3·29-s − 273. i·31-s − 492. i·33-s + 1.54e3·37-s + ⋯
L(s)  = 1  + 1.79i·3-s + 0.313i·5-s − 2.21·9-s − 0.252·11-s − 0.960i·13-s − 0.561·15-s − 1.19i·17-s + 0.731i·19-s − 0.402·23-s + 0.901·25-s − 2.17i·27-s − 1.20·29-s − 0.284i·31-s − 0.452i·33-s + 1.12·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(81.0420\)
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: \((2,\ 784,\ (\ :2),\ 0.912 - 0.409i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.423021433\)
\(L(\frac12)\) \(\approx\) \(1.423021433\)
\(L(3)\) 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 \( 1 - 16.1iT - 81T^{2} \)
5 \( 1 - 7.83iT - 625T^{2} \)
11 \( 1 + 30.5T + 1.46e4T^{2} \)
13 \( 1 + 162. iT - 2.85e4T^{2} \)
17 \( 1 + 345. iT - 8.35e4T^{2} \)
19 \( 1 - 263. iT - 1.30e5T^{2} \)
23 \( 1 + 213.T + 2.79e5T^{2} \)
29 \( 1 + 1.01e3T + 7.07e5T^{2} \)
31 \( 1 + 273. iT - 9.23e5T^{2} \)
37 \( 1 - 1.54e3T + 1.87e6T^{2} \)
41 \( 1 + 531. iT - 2.82e6T^{2} \)
43 \( 1 - 1.47e3T + 3.41e6T^{2} \)
47 \( 1 + 2.38e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.20e3T + 7.89e6T^{2} \)
59 \( 1 - 3.11e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.94e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.91e3T + 2.01e7T^{2} \)
71 \( 1 + 8.58e3T + 2.54e7T^{2} \)
73 \( 1 + 359. iT - 2.83e7T^{2} \)
79 \( 1 + 2.08e3T + 3.89e7T^{2} \)
83 \( 1 + 1.05e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.04e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.40e3iT - 8.85e7T^{2} \)
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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

−9.859252917160547832339234375860, −9.180781027257369157010104119010, −8.271213444244360439938508707900, −7.29294177642518252836693031049, −5.84785118022162715458251403429, −5.27147168230291258053823334884, −4.28890326808742996637862117900, −3.40549070194627006073851302713, −2.55930230175888973018217738306, −0.40137172429730618003178017692, 0.883052000407748912940115904081, 1.79169584790346343064145247780, 2.70777318760967853082535927348, 4.18620670568160507291277335068, 5.52064913080887141691461188886, 6.34234517928668154496120705894, 7.06001356203688608844313914138, 7.85365919337899629220781277324, 8.605027676357942128723137217874, 9.373955148307853725212957010119

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