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Properties

Label 2-728-728.69-c0-0-3
Degree $2$
Conductor $728$
Sign $0.969 + 0.246i$
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  + (0.866 + 0.5i)2-s + (0.965 − 1.67i)3-s + (0.499 + 0.866i)4-s + 0.517i·5-s + (1.67 − 0.965i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−1.36 − 2.36i)9-s + (−0.258 + 0.448i)10-s + 1.93·12-s + (0.258 − 0.965i)13-s − 0.999·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s − 2.73i·18-s + (−1.22 + 0.707i)19-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.965 − 1.67i)3-s + (0.499 + 0.866i)4-s + 0.517i·5-s + (1.67 − 0.965i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−1.36 − 2.36i)9-s + (−0.258 + 0.448i)10-s + 1.93·12-s + (0.258 − 0.965i)13-s − 0.999·14-s + (0.866 + 0.499i)15-s + (−0.5 + 0.866i)16-s − 2.73i·18-s + (−1.22 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(0.363319\)
Root analytic conductor: \(0.602759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :0),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.803763138\)
\(L(\frac12)\) \(\approx\) \(1.803763138\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.258 + 0.965i)T \)
good3 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - 0.517iT - T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−10.78443318983488279097313950765, −9.389253093657897636497418277815, −8.366927810097646315468160348311, −7.896306323410220584185004130274, −6.90188359749276391378708887612, −6.36495098158021418534696989743, −5.63868495808604801257022719206, −3.65615974826060447246056233463, −2.99181115328632732216471201877, −2.05324916624607817725799714606, 2.32584146852629699217200981248, 3.31330026994784052235563321164, 4.32701365998439069473180826403, 4.55232188615521658512712138812, 5.88140138068269180630242901829, 6.98496457691072270427230863832, 8.530939143468166282346698472940, 9.127322892966332610858729419020, 9.902627592855212708622152153986, 10.56863233287718506204765561101

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