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Properties

Label 2-728-56.27-c1-0-21
Degree $2$
Conductor $728$
Sign $-0.754 - 0.656i$
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  + (−1.29 − 0.562i)2-s + 3.37i·3-s + (1.36 + 1.45i)4-s + 0.523·5-s + (1.89 − 4.37i)6-s + (2.30 − 1.29i)7-s + (−0.954 − 2.66i)8-s − 8.38·9-s + (−0.679 − 0.294i)10-s + 1.78·11-s + (−4.92 + 4.61i)12-s + 13-s + (−3.72 + 0.377i)14-s + 1.76i·15-s + (−0.258 + 3.99i)16-s + 5.95i·17-s + ⋯
L(s)  = 1  + (−0.917 − 0.397i)2-s + 1.94i·3-s + (0.683 + 0.729i)4-s + 0.234·5-s + (0.774 − 1.78i)6-s + (0.872 − 0.488i)7-s + (−0.337 − 0.941i)8-s − 2.79·9-s + (−0.214 − 0.0931i)10-s + 0.536·11-s + (−1.42 + 1.33i)12-s + 0.277·13-s + (−0.994 + 0.100i)14-s + 0.456i·15-s + (−0.0645 + 0.997i)16-s + 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.754 - 0.656i$
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: \((2,\ 728,\ (\ :1/2),\ -0.754 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.334055 + 0.892126i\)
\(L(\frac12)\) \(\approx\) \(0.334055 + 0.892126i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.562i)T \)
7 \( 1 + (-2.30 + 1.29i)T \)
13 \( 1 - T \)
good3 \( 1 - 3.37iT - 3T^{2} \)
5 \( 1 - 0.523T + 5T^{2} \)
11 \( 1 - 1.78T + 11T^{2} \)
17 \( 1 - 5.95iT - 17T^{2} \)
19 \( 1 - 1.23iT - 19T^{2} \)
23 \( 1 - 6.84iT - 23T^{2} \)
29 \( 1 - 8.12iT - 29T^{2} \)
31 \( 1 + 2.22T + 31T^{2} \)
37 \( 1 + 8.61iT - 37T^{2} \)
41 \( 1 - 5.57iT - 41T^{2} \)
43 \( 1 + 4.02T + 43T^{2} \)
47 \( 1 + 5.78T + 47T^{2} \)
53 \( 1 - 1.99iT - 53T^{2} \)
59 \( 1 + 3.68iT - 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 3.23T + 67T^{2} \)
71 \( 1 - 0.681iT - 71T^{2} \)
73 \( 1 - 2.59iT - 73T^{2} \)
79 \( 1 + 6.79iT - 79T^{2} \)
83 \( 1 + 9.47iT - 83T^{2} \)
89 \( 1 + 1.03iT - 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−10.52656807820387128577381208831, −9.980332431516664611881584018862, −9.161492965955877004884176719887, −8.530701834116668292798816343552, −7.68233206400123891437803973531, −6.18393992764448811141545517381, −5.19017822540755352576002620029, −3.92115993250340006082661830171, −3.52759451444018632311415522966, −1.75794343652850009730548865415, 0.66724463949675644864547567942, 1.87103554143752395484047552972, 2.61945371436118391062461848466, 5.10808025704078730144008992626, 6.05408198166534102173735658905, 6.71783884220573898912970998817, 7.50011772834773109913826632202, 8.285078064272346600687068484897, 8.771824017281458048167217944604, 9.837502086113228019893892080076

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