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Properties

Label 2-1568-224.173-c0-0-0
Degree $2$
Conductor $1568$
Sign $0.610 - 0.792i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.12 − 1.46i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 + 0.707i)22-s + (1.36 − 0.366i)23-s + (−0.965 − 0.258i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s + (−0.707 + 0.707i)36-s + (−0.0999 + 0.758i)37-s + (−0.292 − 0.707i)43-s + (−0.241 + 1.83i)44-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (1.12 − 1.46i)11-s + (0.500 − 0.866i)16-s + (0.499 + 0.866i)18-s + (1.70 + 0.707i)22-s + (1.36 − 0.366i)23-s + (−0.965 − 0.258i)25-s + (−0.707 + 1.70i)29-s + (0.965 + 0.258i)32-s + (−0.707 + 0.707i)36-s + (−0.0999 + 0.758i)37-s + (−0.292 − 0.707i)43-s + (−0.241 + 1.83i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.610 - 0.792i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.610 - 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280957588\)
\(L(\frac12)\) \(\approx\) \(1.280957588\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (-0.965 + 0.258i)T^{2} \)
5 \( 1 + (0.965 + 0.258i)T^{2} \)
11 \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.258 + 0.965i)T^{2} \)
23 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.0999 - 0.758i)T + (-0.965 - 0.258i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \)
59 \( 1 + (-0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.758 - 0.0999i)T + (0.965 - 0.258i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
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

−9.341255210860866679576329769159, −8.937677909035605305901842537539, −8.113961794057146236712684611177, −7.11510345920967034734913939759, −6.60326945495925836457551768876, −5.78304605084551903314748723365, −4.86205494055028370454578404327, −3.87850318134142175550693680608, −3.22950628708172012774376965815, −1.20258403451416158438986234569, 1.43890372712622404381595271969, 2.22596918028077293939360721292, 3.65832458456491740958238174288, 4.32091839715597638296715882756, 5.04508632563340569865273566480, 6.19025515468313009349656577204, 7.14457367000993295293657947190, 7.903583966486814698170445921189, 9.235612918639542493764515952061, 9.528095673270649420172139959514

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