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Properties

Label 2-1568-56.11-c0-0-2
Degree $2$
Conductor $1568$
Sign $0.0725 + 0.997i$
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  + (0.707 − 1.22i)3-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + 1.41·41-s + (−0.999 − 1.73i)51-s − 2·57-s + (−0.707 + 1.22i)59-s + (−1 + 1.73i)67-s + (−0.707 + 1.22i)73-s + (0.707 + 1.22i)75-s + (0.499 − 0.866i)81-s + 1.41·83-s + (−0.707 − 1.22i)89-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)3-s + (−0.499 − 0.866i)9-s + (0.707 − 1.22i)17-s + (−0.707 − 1.22i)19-s + (−0.5 + 0.866i)25-s + 1.41·41-s + (−0.999 − 1.73i)51-s − 2·57-s + (−0.707 + 1.22i)59-s + (−1 + 1.73i)67-s + (−0.707 + 1.22i)73-s + (0.707 + 1.22i)75-s + (0.499 − 0.866i)81-s + 1.41·83-s + (−0.707 − 1.22i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.0725 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380486867\)
\(L(\frac12)\) \(\approx\) \(1.380486867\)
\(L(1)\) 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 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.157798156523808286650892453451, −8.660605504012648309813309161355, −7.53992890417129484905562244267, −7.35063504193994252362671774154, −6.41080785281774541841876592879, −5.43722138909765229489834439863, −4.35125661697436964307967823993, −3.04325926379415350775855125820, −2.36511757652262827163282753088, −1.10601424514485362009228973435, 1.86858885853818821176704884330, 3.13352330154070857860618869330, 3.92114328232216986708507225215, 4.52900856175681329269914188302, 5.70244875009684826986498802205, 6.40499314013044653506388457593, 7.84689217463132995145925867823, 8.226287700604041596649903023215, 9.138573588130012159187472863741, 9.799173086239167911316777520391

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