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Properties

Label 2-728-8.5-c1-0-4
Degree $2$
Conductor $728$
Sign $-0.0316 + 0.999i$
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  + (−0.720 + 1.21i)2-s + 2.66i·3-s + (−0.963 − 1.75i)4-s + 2.77i·5-s + (−3.24 − 1.91i)6-s − 7-s + (2.82 + 0.0896i)8-s − 4.09·9-s + (−3.37 − 1.99i)10-s + 2.30i·11-s + (4.66 − 2.56i)12-s − i·13-s + (0.720 − 1.21i)14-s − 7.38·15-s + (−2.14 + 3.37i)16-s − 6.69·17-s + ⋯
L(s)  = 1  + (−0.509 + 0.860i)2-s + 1.53i·3-s + (−0.481 − 0.876i)4-s + 1.24i·5-s + (−1.32 − 0.782i)6-s − 0.377·7-s + (0.999 + 0.0316i)8-s − 1.36·9-s + (−1.06 − 0.631i)10-s + 0.695i·11-s + (1.34 − 0.740i)12-s − 0.277i·13-s + (0.192 − 0.325i)14-s − 1.90·15-s + (−0.536 + 0.844i)16-s − 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0316 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0316 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.0316 + 0.999i$
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: \((2,\ 728,\ (\ :1/2),\ -0.0316 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448666 - 0.463117i\)
\(L(\frac12)\) \(\approx\) \(0.448666 - 0.463117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.720 - 1.21i)T \)
7 \( 1 + T \)
13 \( 1 + iT \)
good3 \( 1 - 2.66iT - 3T^{2} \)
5 \( 1 - 2.77iT - 5T^{2} \)
11 \( 1 - 2.30iT - 11T^{2} \)
17 \( 1 + 6.69T + 17T^{2} \)
19 \( 1 + 2.40iT - 19T^{2} \)
23 \( 1 - 5.02T + 23T^{2} \)
29 \( 1 - 4.22iT - 29T^{2} \)
31 \( 1 + 6.52T + 31T^{2} \)
37 \( 1 + 7.64iT - 37T^{2} \)
41 \( 1 + 1.66T + 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 + 0.579iT - 53T^{2} \)
59 \( 1 - 10.2iT - 59T^{2} \)
61 \( 1 - 7.92iT - 61T^{2} \)
67 \( 1 + 6.80iT - 67T^{2} \)
71 \( 1 - 7.75T + 71T^{2} \)
73 \( 1 + 5.71T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 0.262iT - 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−10.92435558244894969882413043443, −10.13347009494807516262376978059, −9.205537280530645178759103009597, −8.883906218663343648507733961318, −7.30013255032283337389521799438, −6.85104947385383207514445031568, −5.71663344054454279315410372678, −4.75476753819466220336343954631, −3.86921940629653005875680099710, −2.57983669295091150985408068382, 0.40339681764492041515788544571, 1.48182192802350843761637781079, 2.52345162044110351999803994466, 3.95924167838012573616005125392, 5.18309081753246790477516277657, 6.44433787778470579719516356690, 7.29674763438832992353721548408, 8.321043290083674204071552751840, 8.755596145586211555186487308329, 9.519925999810569349363036106896

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