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Properties

Label 2-728-56.27-c1-0-5
Degree $2$
Conductor $728$
Sign $-0.999 + 0.0115i$
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  + (−0.172 − 1.40i)2-s + 3.03i·3-s + (−1.94 + 0.483i)4-s + 0.163·5-s + (4.26 − 0.522i)6-s + (−0.919 + 2.48i)7-s + (1.01 + 2.64i)8-s − 6.21·9-s + (−0.0282 − 0.230i)10-s − 4.50·11-s + (−1.46 − 5.89i)12-s + 13-s + (3.64 + 0.862i)14-s + 0.497i·15-s + (3.53 − 1.87i)16-s − 3.02i·17-s + ⋯
L(s)  = 1  + (−0.121 − 0.992i)2-s + 1.75i·3-s + (−0.970 + 0.241i)4-s + 0.0732·5-s + (1.73 − 0.213i)6-s + (−0.347 + 0.937i)7-s + (0.358 + 0.933i)8-s − 2.07·9-s + (−0.00893 − 0.0727i)10-s − 1.35·11-s + (−0.423 − 1.70i)12-s + 0.277·13-s + (0.973 + 0.230i)14-s + 0.128i·15-s + (0.883 − 0.469i)16-s − 0.733i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0115i$
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: \((2,\ 728,\ (\ :1/2),\ -0.999 + 0.0115i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00203887 - 0.352005i\)
\(L(\frac12)\) \(\approx\) \(0.00203887 - 0.352005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.172 + 1.40i)T \)
7 \( 1 + (0.919 - 2.48i)T \)
13 \( 1 - T \)
good3 \( 1 - 3.03iT - 3T^{2} \)
5 \( 1 - 0.163T + 5T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
17 \( 1 + 3.02iT - 17T^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 - 4.64iT - 23T^{2} \)
29 \( 1 + 2.17iT - 29T^{2} \)
31 \( 1 - 8.62T + 31T^{2} \)
37 \( 1 + 9.21iT - 37T^{2} \)
41 \( 1 - 6.42iT - 41T^{2} \)
43 \( 1 + 7.59T + 43T^{2} \)
47 \( 1 + 2.11T + 47T^{2} \)
53 \( 1 - 2.63iT - 53T^{2} \)
59 \( 1 - 13.3iT - 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 0.950iT - 71T^{2} \)
73 \( 1 - 0.307iT - 73T^{2} \)
79 \( 1 - 13.7iT - 79T^{2} \)
83 \( 1 - 1.24iT - 83T^{2} \)
89 \( 1 - 14.7iT - 89T^{2} \)
97 \( 1 + 7.58iT - 97T^{2} \)
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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

−10.71166273802645235674955093074, −9.922139180724897976480903498235, −9.443778740268592850478549255510, −8.734930134131271687753653268141, −7.82120195756816816256328415303, −5.83653141042706528959805670390, −5.14438166575038685126562374065, −4.37481660527341540811417118851, −3.16077810754100757018738334924, −2.56054578320192019404468446356, 0.18639047892101724612361073473, 1.60481610572790130707583390295, 3.25679552393468688023774932362, 4.75941902910721890853041454646, 6.03448341632567432213527241825, 6.44579600917997379046168037611, 7.40808845091612103040433739897, 8.024138373436857710141152972749, 8.479990381023896447682722379850, 10.01076105053546308271257935492

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