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Properties

Label 2-728-728.675-c1-0-31
Degree $2$
Conductor $728$
Sign $-0.620 + 0.784i$
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  + (−0.707 − 1.22i)2-s + (−2.41 − 1.39i)3-s + (−0.999 + 1.73i)4-s + (−2.74 + 1.58i)5-s + 3.94i·6-s + (0.710 − 2.54i)7-s + 2.82·8-s + (2.39 + 4.15i)9-s + (3.88 + 2.24i)10-s + (4.83 − 2.79i)12-s + 3.60i·13-s + (−3.62 + 0.931i)14-s + 8.84·15-s + (−2.00 − 3.46i)16-s + (5.61 + 3.24i)17-s + (3.38 − 5.87i)18-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s + (−1.39 − 0.805i)3-s + (−0.499 + 0.866i)4-s + (−1.22 + 0.708i)5-s + 1.61i·6-s + (0.268 − 0.963i)7-s + 0.999·8-s + (0.798 + 1.38i)9-s + (1.22 + 0.708i)10-s + (1.39 − 0.805i)12-s + 0.999i·13-s + (−0.968 + 0.248i)14-s + 2.28·15-s + (−0.500 − 0.866i)16-s + (1.36 + 0.786i)17-s + (0.798 − 1.38i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.620 + 0.784i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.620 + 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175380 - 0.362174i\)
\(L(\frac12)\) \(\approx\) \(0.175380 - 0.362174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
7 \( 1 + (-0.710 + 2.54i)T \)
13 \( 1 - 3.60iT \)
good3 \( 1 + (2.41 + 1.39i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.74 - 1.58i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.61 - 3.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (9.48 + 5.47i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.06 + 8.76i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13.1T + 43T^{2} \)
47 \( 1 + (2.34 - 1.35i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.68041190942427509803948030651, −9.486613400494767032136238800714, −8.063669915299760872325666110571, −7.39878825323980076821745352622, −6.98897136947671177415222251097, −5.64201355244744292916018676958, −4.26351809838524161211604088700, −3.61344301492177075435747829750, −1.75774673889826632871080382800, −0.45829451794364199343323664764, 0.849486795646886939645714661023, 3.62697820487122897670744888853, 4.94627667959251472910048633083, 5.20143329358368631194174233318, 6.03255250347620464395119594944, 7.31697855354850006980722452295, 8.091607957025637178421662561060, 8.943676678921623024166771633497, 9.799579724458806372625686435682, 10.67375261011096382397958325951

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