Properties

Degree $2$
Conductor $1568$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s − 0.999·9-s − 2·11-s − 2.82·13-s − 4.24·17-s − 4.24·19-s − 8·23-s − 5·25-s − 5.65·27-s + 6·29-s + 8.48·31-s − 2.82·33-s − 2·37-s − 4.00·39-s + 4.24·41-s − 6·43-s − 2.82·47-s − 6·51-s + 6·53-s − 6·57-s + 12.7·59-s + 5.65·61-s − 12·67-s − 11.3·69-s + 4·71-s + 1.41·73-s − 7.07·75-s + ⋯
L(s)  = 1  + 0.816·3-s − 0.333·9-s − 0.603·11-s − 0.784·13-s − 1.02·17-s − 0.973·19-s − 1.66·23-s − 25-s − 1.08·27-s + 1.11·29-s + 1.52·31-s − 0.492·33-s − 0.328·37-s − 0.640·39-s + 0.662·41-s − 0.914·43-s − 0.412·47-s − 0.840·51-s + 0.824·53-s − 0.794·57-s + 1.65·59-s + 0.724·61-s − 1.46·67-s − 1.36·69-s + 0.474·71-s + 0.165·73-s − 0.816·75-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1568} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + 4.24T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−8.796411830139782650515102904746, −8.332783511928667936434927338805, −7.66893629872196205128658087659, −6.62059559230917785167409095120, −5.82698469204577047203088655363, −4.70027223285448322345911122155, −3.90648630375322893589313542854, −2.65870680996730301452400797456, −2.12947067911120445145131286293, 0, 2.12947067911120445145131286293, 2.65870680996730301452400797456, 3.90648630375322893589313542854, 4.70027223285448322345911122155, 5.82698469204577047203088655363, 6.62059559230917785167409095120, 7.66893629872196205128658087659, 8.332783511928667936434927338805, 8.796411830139782650515102904746

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