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Properties

Label 2-728-728.571-c0-0-1
Degree $2$
Conductor $728$
Sign $0.400 - 0.916i$
Analytic cond. $0.363319$
Root an. cond. $0.602759$
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  + (−0.5 − 0.866i)2-s + (−0.766 + 1.32i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + (0.766 − 0.642i)7-s + 0.999·8-s + (−0.673 − 1.16i)9-s + (0.939 − 1.62i)10-s + (−0.766 − 1.32i)12-s + 13-s + (−0.939 − 0.342i)14-s − 2.87·15-s + (−0.5 − 0.866i)16-s + (−0.173 + 0.300i)17-s + (−0.673 + 1.16i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 + 1.32i)3-s + (−0.499 + 0.866i)4-s + (0.939 + 1.62i)5-s + 1.53·6-s + (0.766 − 0.642i)7-s + 0.999·8-s + (−0.673 − 1.16i)9-s + (0.939 − 1.62i)10-s + (−0.766 − 1.32i)12-s + 13-s + (−0.939 − 0.342i)14-s − 2.87·15-s + (−0.5 − 0.866i)16-s + (−0.173 + 0.300i)17-s + (−0.673 + 1.16i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7145941770\)
\(L(\frac12)\) \(\approx\) \(0.7145941770\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 - T \)
good3 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.87T + T^{2} \)
47 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 0.347T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−10.76652306429724869938642551776, −10.15876021358364465108240169645, −9.592957165843643450678748654411, −8.480944902670177016379999090219, −7.28571964248952547031922588487, −6.27055106705662348541948202154, −5.23598991571887818740669365644, −4.06218805554637986092414346063, −3.34649718180602442471678241961, −1.92580869753063113066288150171, 1.18013923071166045748001467308, 1.79876621633891232760747008070, 4.76771858041318593856184519753, 5.32102613311105753897672002550, 6.07639122919980987448816085856, 6.75489963167013748081315921839, 8.083546361754245083883054230087, 8.467815631710911945227530890184, 9.229550266598375253458869559798, 10.29404243469278849743725374537

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