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Properties

Label 2-728-91.16-c1-0-14
Degree $2$
Conductor $728$
Sign $0.987 + 0.156i$
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  + (1.02 − 1.77i)3-s + (−0.859 + 1.48i)5-s + (−0.5 + 2.59i)7-s + (−0.594 − 1.02i)9-s + (1.45 − 2.51i)11-s + (3.44 + 1.06i)13-s + (1.75 + 3.04i)15-s + 2.57·17-s + (−2.47 − 4.29i)19-s + (4.09 + 3.54i)21-s + 5.70·23-s + (1.02 + 1.77i)25-s + 3.70·27-s + (1.82 + 3.16i)29-s + (2.66 + 4.61i)31-s + ⋯
L(s)  = 1  + (0.590 − 1.02i)3-s + (−0.384 + 0.665i)5-s + (−0.188 + 0.981i)7-s + (−0.198 − 0.343i)9-s + (0.438 − 0.759i)11-s + (0.955 + 0.294i)13-s + (0.454 + 0.786i)15-s + 0.625·17-s + (−0.568 − 0.984i)19-s + (0.893 + 0.773i)21-s + 1.18·23-s + (0.204 + 0.354i)25-s + 0.713·27-s + (0.339 + 0.588i)29-s + (0.478 + 0.828i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84661 - 0.145271i\)
\(L(\frac12)\) \(\approx\) \(1.84661 - 0.145271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
13 \( 1 + (-3.44 - 1.06i)T \)
good3 \( 1 + (-1.02 + 1.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.859 - 1.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.45 + 2.51i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 + (2.47 + 4.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 + (-1.82 - 3.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.66 - 4.61i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 + (-3.76 - 6.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.42 + 4.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.39 + 4.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.563 - 0.976i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.87T + 59T^{2} \)
61 \( 1 + (7.44 + 12.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.882 - 1.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.53 - 2.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.56 + 9.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.61 + 2.80i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.52T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + (4.39 - 7.60i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60820057921814842509360748942, −9.019177810517640619877708847551, −8.765105063199321161129079683757, −7.77571578615848514351562936446, −6.81269709412316679680754950283, −6.31613276949490974384877838678, −5.03245771496723387898034162060, −3.39363571364517480009948698358, −2.75669070816852920279907684751, −1.35198541556628765080714919040, 1.14646782952004094109255020087, 3.11854572946414819187303991888, 4.15250462448342586771329693227, 4.43688130950090900578537117066, 5.88150131592173786949886761421, 7.04844249124577664704623600655, 8.039497502085558734805746394543, 8.778509031727308508992383765204, 9.599478611728801179610899052219, 10.29749571972761276108211985176

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