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Properties

Label 2-728-56.27-c1-0-44
Degree $2$
Conductor $728$
Sign $0.988 + 0.153i$
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  + (−0.635 − 1.26i)2-s + 1.40i·3-s + (−1.19 + 1.60i)4-s + 3.11·5-s + (1.77 − 0.891i)6-s + (2.64 − 0.0479i)7-s + (2.78 + 0.485i)8-s + 1.02·9-s + (−1.97 − 3.93i)10-s − 0.711·11-s + (−2.25 − 1.67i)12-s + 13-s + (−1.74 − 3.31i)14-s + 4.37i·15-s + (−1.15 − 3.82i)16-s − 0.128i·17-s + ⋯
L(s)  = 1  + (−0.449 − 0.893i)2-s + 0.810i·3-s + (−0.596 + 0.802i)4-s + 1.39·5-s + (0.723 − 0.364i)6-s + (0.999 − 0.0181i)7-s + (0.985 + 0.171i)8-s + 0.343·9-s + (−0.625 − 1.24i)10-s − 0.214·11-s + (−0.650 − 0.483i)12-s + 0.277·13-s + (−0.465 − 0.885i)14-s + 1.12i·15-s + (−0.289 − 0.957i)16-s − 0.0310i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.988 + 0.153i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69223 - 0.131020i\)
\(L(\frac12)\) \(\approx\) \(1.69223 - 0.131020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.635 + 1.26i)T \)
7 \( 1 + (-2.64 + 0.0479i)T \)
13 \( 1 - T \)
good3 \( 1 - 1.40iT - 3T^{2} \)
5 \( 1 - 3.11T + 5T^{2} \)
11 \( 1 + 0.711T + 11T^{2} \)
17 \( 1 + 0.128iT - 17T^{2} \)
19 \( 1 + 5.49iT - 19T^{2} \)
23 \( 1 - 5.12iT - 23T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 8.15iT - 37T^{2} \)
41 \( 1 + 0.100iT - 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + 2.81T + 47T^{2} \)
53 \( 1 + 8.53iT - 53T^{2} \)
59 \( 1 + 7.38iT - 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 14.5T + 67T^{2} \)
71 \( 1 - 8.43iT - 71T^{2} \)
73 \( 1 + 8.62iT - 73T^{2} \)
79 \( 1 - 16.3iT - 79T^{2} \)
83 \( 1 + 4.06iT - 83T^{2} \)
89 \( 1 - 2.25iT - 89T^{2} \)
97 \( 1 - 5.13iT - 97T^{2} \)
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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

−10.32128891351991935880085631211, −9.554255154762058391203395939307, −9.151561308300013576643119677676, −8.077398832162350366971149980310, −7.03735854023169567718478687558, −5.52259507041689550489360499568, −4.83956061766999510086533102571, −3.80953082137758215725618791298, −2.42415507323884816523526320130, −1.44817640454258393803791237819, 1.34132448403182991316103897792, 2.05410617923692820792729186959, 4.28542857693724741500168623498, 5.50675312765070196914873712143, 5.95423511497639713895445727838, 7.04225924021755817987532774261, 7.66370828790726998210610401109, 8.636962498291762484914825108883, 9.332720939500369112520945475237, 10.40605923157983865083110743262

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