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Properties

Label 2-728-104.101-c1-0-11
Degree $2$
Conductor $728$
Sign $-0.965 - 0.261i$
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  + (1.30 + 0.555i)2-s + (−0.563 + 0.325i)3-s + (1.38 + 1.44i)4-s − 2.56·5-s + (−0.913 + 0.109i)6-s + (0.866 + 0.5i)7-s + (0.993 + 2.64i)8-s + (−1.28 + 2.23i)9-s + (−3.32 − 1.42i)10-s + (−2.93 − 5.08i)11-s + (−1.24 − 0.364i)12-s + (−2.06 + 2.95i)13-s + (0.848 + 1.13i)14-s + (1.44 − 0.833i)15-s + (−0.179 + 3.99i)16-s + (−0.197 + 0.341i)17-s + ⋯
L(s)  = 1  + (0.919 + 0.393i)2-s + (−0.325 + 0.187i)3-s + (0.691 + 0.722i)4-s − 1.14·5-s + (−0.373 + 0.0448i)6-s + (0.327 + 0.188i)7-s + (0.351 + 0.936i)8-s + (−0.429 + 0.743i)9-s + (−1.05 − 0.450i)10-s + (−0.884 − 1.53i)11-s + (−0.360 − 0.105i)12-s + (−0.573 + 0.819i)13-s + (0.226 + 0.302i)14-s + (0.372 − 0.215i)15-s + (−0.0448 + 0.998i)16-s + (−0.0478 + 0.0828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.965 - 0.261i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.965 - 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.150396 + 1.12980i\)
\(L(\frac12)\) \(\approx\) \(0.150396 + 1.12980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.555i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (2.06 - 2.95i)T \)
good3 \( 1 + (0.563 - 0.325i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
11 \( 1 + (2.93 + 5.08i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.197 - 0.341i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.11 - 7.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.24 - 3.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.92 + 2.26i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.34iT - 31T^{2} \)
37 \( 1 + (2.90 + 5.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.40 - 4.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.49 - 4.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.44iT - 47T^{2} \)
53 \( 1 + 0.739iT - 53T^{2} \)
59 \( 1 + (6.84 - 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.02 - 1.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.87 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.3 - 6.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.41iT - 73T^{2} \)
79 \( 1 - 2.10T + 79T^{2} \)
83 \( 1 - 3.25T + 83T^{2} \)
89 \( 1 + (-7.13 + 4.11i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.67 + 0.965i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.16141769675854430286260361056, −10.30717495399846472117003351030, −8.549123546384459924493448446063, −8.105804979685397008519399735223, −7.39055553928613589296624893607, −6.07783254237821151380748039670, −5.42791826560996444039228268668, −4.42622725519113707574178418375, −3.58081411919286585957517529275, −2.36140225890884409019266287003, 0.42363717428719944652164343372, 2.37643441512388277208522841939, 3.41464565351722979214410200386, 4.76701306304250377073061340001, 4.94840498016245527578534427027, 6.57370210715904759552317938919, 7.13251860755257281615589467791, 8.048659902217698467798156264677, 9.268748784715061500706469096147, 10.49657478872941868653836822522

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