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Properties

Label 2-728-8.5-c1-0-14
Degree $2$
Conductor $728$
Sign $0.891 + 0.452i$
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  + (0.220 − 1.39i)2-s − 1.54i·3-s + (−1.90 − 0.615i)4-s + 3.27i·5-s + (−2.16 − 0.341i)6-s − 7-s + (−1.27 + 2.52i)8-s + 0.600·9-s + (4.57 + 0.721i)10-s + 2.39i·11-s + (−0.953 + 2.94i)12-s + i·13-s + (−0.220 + 1.39i)14-s + 5.07·15-s + (3.24 + 2.34i)16-s + 2.51·17-s + ⋯
L(s)  = 1  + (0.155 − 0.987i)2-s − 0.894i·3-s + (−0.951 − 0.307i)4-s + 1.46i·5-s + (−0.883 − 0.139i)6-s − 0.377·7-s + (−0.452 + 0.891i)8-s + 0.200·9-s + (1.44 + 0.228i)10-s + 0.723i·11-s + (−0.275 + 0.850i)12-s + 0.277i·13-s + (−0.0589 + 0.373i)14-s + 1.30·15-s + (0.810 + 0.585i)16-s + 0.609·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.891 + 0.452i$
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: \((2,\ 728,\ (\ :1/2),\ 0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34899 - 0.322565i\)
\(L(\frac12)\) \(\approx\) \(1.34899 - 0.322565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.220 + 1.39i)T \)
7 \( 1 + T \)
13 \( 1 - iT \)
good3 \( 1 + 1.54iT - 3T^{2} \)
5 \( 1 - 3.27iT - 5T^{2} \)
11 \( 1 - 2.39iT - 11T^{2} \)
17 \( 1 - 2.51T + 17T^{2} \)
19 \( 1 - 1.28iT - 19T^{2} \)
23 \( 1 - 6.33T + 23T^{2} \)
29 \( 1 - 3.38iT - 29T^{2} \)
31 \( 1 - 1.78T + 31T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + 3.81T + 41T^{2} \)
43 \( 1 + 5.92iT - 43T^{2} \)
47 \( 1 - 3.49T + 47T^{2} \)
53 \( 1 - 12.6iT - 53T^{2} \)
59 \( 1 - 9.43iT - 59T^{2} \)
61 \( 1 + 6.93iT - 61T^{2} \)
67 \( 1 + 1.35iT - 67T^{2} \)
71 \( 1 - 4.37T + 71T^{2} \)
73 \( 1 + 5.91T + 73T^{2} \)
79 \( 1 + 4.84T + 79T^{2} \)
83 \( 1 + 9.94iT - 83T^{2} \)
89 \( 1 - 0.0174T + 89T^{2} \)
97 \( 1 + 5.76T + 97T^{2} \)
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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

−10.31476046287195854948992428690, −9.898468884116221140371212078168, −8.741882907609383692517524911594, −7.50362290601879540734839704168, −6.92359552183832710531503335919, −6.02697388998388842419543997439, −4.66434345663144777142380081576, −3.39733105928821420920163529648, −2.60025813545294949155971784683, −1.41526871021225507928911174490, 0.78530524626327563962633186045, 3.37073673773088667543906910209, 4.29265603667248354087497390915, 5.10973371941695530808745583213, 5.70506261257356971659351955987, 6.94580407969196881119052931411, 8.019862987374512285780699209797, 8.799833928942580057957788599888, 9.376427951339628180169849066760, 10.06804891166825039574503418115

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