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Properties

Label 2-728-8.5-c1-0-23
Degree $2$
Conductor $728$
Sign $0.998 - 0.0516i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 − 0.685i)2-s + 0.302i·3-s + (1.05 + 1.69i)4-s + 1.42i·5-s + (0.207 − 0.373i)6-s − 7-s + (−0.146 − 2.82i)8-s + 2.90·9-s + (0.979 − 1.76i)10-s − 2.11i·11-s + (−0.512 + 0.320i)12-s − i·13-s + (1.23 + 0.685i)14-s − 0.431·15-s + (−1.75 + 3.59i)16-s + 1.07·17-s + ⋯
L(s)  = 1  + (−0.874 − 0.485i)2-s + 0.174i·3-s + (0.529 + 0.848i)4-s + 0.638i·5-s + (0.0846 − 0.152i)6-s − 0.377·7-s + (−0.0516 − 0.998i)8-s + 0.969·9-s + (0.309 − 0.558i)10-s − 0.636i·11-s + (−0.148 + 0.0924i)12-s − 0.277i·13-s + (0.330 + 0.183i)14-s − 0.111·15-s + (−0.439 + 0.898i)16-s + 0.260·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.998 - 0.0516i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.998 - 0.0516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06064 + 0.0274055i\)
\(L(\frac12)\) \(\approx\) \(1.06064 + 0.0274055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.685i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 0.302iT - 3T^{2} \)
5 \( 1 - 1.42iT - 5T^{2} \)
11 \( 1 + 2.11iT - 11T^{2} \)
17 \( 1 - 1.07T + 17T^{2} \)
19 \( 1 + 3.52iT - 19T^{2} \)
23 \( 1 - 5.47T + 23T^{2} \)
29 \( 1 - 3.52iT - 29T^{2} \)
31 \( 1 + 3.26T + 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 - 9.67T + 41T^{2} \)
43 \( 1 - 9.02iT - 43T^{2} \)
47 \( 1 - 4.79T + 47T^{2} \)
53 \( 1 + 0.906iT - 53T^{2} \)
59 \( 1 + 6.88iT - 59T^{2} \)
61 \( 1 + 5.18iT - 61T^{2} \)
67 \( 1 - 5.46iT - 67T^{2} \)
71 \( 1 - 0.895T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + 9.02iT - 83T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 4.26T + 97T^{2} \)
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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

−10.52297815239691701564374041775, −9.548268759939216469065509030990, −8.950265384373717283441315038149, −7.83376829039166366931659444047, −7.03620528444657569714337949528, −6.35386123935785694377822481960, −4.79905335954163140567817346369, −3.47585959028357381034632300528, −2.74995624573155479390262397551, −1.08978492726076710594466014603, 0.985237475905899540889669939809, 2.21485845346884129426264516596, 4.04859856768092449318859604407, 5.15374608312370003212589558241, 6.12780221211711932604903584885, 7.21028009357094327860754636179, 7.58906307102751028824748366607, 8.877373640626133400701624349177, 9.328156045235719640133746103758, 10.22429393169499830929322762713

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