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Properties

Label 2-1568-4.3-c0-0-1
Degree $2$
Conductor $1568$
Sign $0.707 - 0.707i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 5-s − i·11-s + i·15-s + 17-s + i·19-s − i·23-s + i·27-s − i·31-s + 33-s − 37-s + i·47-s + i·51-s − 53-s − i·55-s + ⋯
L(s)  = 1  + i·3-s + 5-s − i·11-s + i·15-s + 17-s + i·19-s − i·23-s + i·27-s − i·31-s + 33-s − 37-s + i·47-s + i·51-s − 53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.378727845\)
\(L(\frac12)\) \(\approx\) \(1.378727845\)
\(L(1)\) 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 \( 1 - iT - T^{2} \)
5 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.844069496557778363775143651290, −9.117489341068918234323084064421, −8.323277945073437730077714014095, −7.38177324660796303016185313565, −6.09364807338881363558493813123, −5.75072775901139941769819133575, −4.73938374713930594684475149310, −3.80310914333514510082957967633, −2.90773159189194057370505319824, −1.52471045353919524105915637814, 1.42672078864925063800081401682, 2.10477350216239990248498809648, 3.32325707759592038979805206487, 4.72610247411982220601576018410, 5.51356197857616680080193634163, 6.43442183342761538884938724620, 7.12261359261468175086451372794, 7.69382019325584655961795644052, 8.737361483558156476068023387667, 9.680990185620814335472449708375

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