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Properties

Label 2-728-56.27-c1-0-9
Degree $2$
Conductor $728$
Sign $-0.272 - 0.962i$
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  + (−0.879 + 1.10i)2-s − 1.92i·3-s + (−0.453 − 1.94i)4-s − 2.16·5-s + (2.13 + 1.69i)6-s + (1.74 + 1.99i)7-s + (2.55 + 1.20i)8-s − 0.716·9-s + (1.90 − 2.40i)10-s − 5.04·11-s + (−3.75 + 0.875i)12-s − 13-s + (−3.73 + 0.175i)14-s + 4.17i·15-s + (−3.58 + 1.76i)16-s − 0.687i·17-s + ⋯
L(s)  = 1  + (−0.621 + 0.783i)2-s − 1.11i·3-s + (−0.226 − 0.973i)4-s − 0.969·5-s + (0.871 + 0.691i)6-s + (0.657 + 0.753i)7-s + (0.903 + 0.427i)8-s − 0.238·9-s + (0.602 − 0.758i)10-s − 1.52·11-s + (−1.08 + 0.252i)12-s − 0.277·13-s + (−0.998 + 0.0467i)14-s + 1.07i·15-s + (−0.896 + 0.442i)16-s − 0.166i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.272 - 0.962i$
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: \((2,\ 728,\ (\ :1/2),\ -0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.313097 + 0.413991i\)
\(L(\frac12)\) \(\approx\) \(0.313097 + 0.413991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.879 - 1.10i)T \)
7 \( 1 + (-1.74 - 1.99i)T \)
13 \( 1 + T \)
good3 \( 1 + 1.92iT - 3T^{2} \)
5 \( 1 + 2.16T + 5T^{2} \)
11 \( 1 + 5.04T + 11T^{2} \)
17 \( 1 + 0.687iT - 17T^{2} \)
19 \( 1 - 1.09iT - 19T^{2} \)
23 \( 1 - 3.80iT - 23T^{2} \)
29 \( 1 - 6.62iT - 29T^{2} \)
31 \( 1 - 8.65T + 31T^{2} \)
37 \( 1 - 9.25iT - 37T^{2} \)
41 \( 1 - 6.30iT - 41T^{2} \)
43 \( 1 + 3.27T + 43T^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 - 2.89iT - 53T^{2} \)
59 \( 1 - 11.6iT - 59T^{2} \)
61 \( 1 + 1.96T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 + 5.07iT - 71T^{2} \)
73 \( 1 + 2.08iT - 73T^{2} \)
79 \( 1 - 0.772iT - 79T^{2} \)
83 \( 1 - 5.41iT - 83T^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 2.48iT - 97T^{2} \)
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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

−10.58373597972305516162243942562, −9.658203704439999552276197716875, −8.318203440289012057093542626195, −8.068125848680567315201490669127, −7.41588270652621141302907985277, −6.52407685504764355132686393953, −5.42619057806703177751876605847, −4.64191208908345333248339949803, −2.70048102153140039854288844008, −1.36079035225726053083128078449, 0.35152070867924025116095917506, 2.41000445263350574469010040339, 3.68927245939636919053781137861, 4.36212534128566933258694178682, 5.09853364778899673203416763914, 7.05456225432691615884664765394, 7.954439873454884786737092389634, 8.326212536037217435835841431383, 9.615564056641862873902948102689, 10.25649676463934081390671315587

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