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Properties

Label 2-728-8.5-c1-0-18
Degree $2$
Conductor $728$
Sign $0.257 - 0.966i$
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  + (−1.40 − 0.122i)2-s + 0.192i·3-s + (1.96 + 0.346i)4-s + 2.98i·5-s + (0.0236 − 0.271i)6-s + 7-s + (−2.73 − 0.729i)8-s + 2.96·9-s + (0.366 − 4.20i)10-s + 1.00i·11-s + (−0.0666 + 0.379i)12-s + i·13-s + (−1.40 − 0.122i)14-s − 0.574·15-s + (3.76 + 1.36i)16-s + 5.01·17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0868i)2-s + 0.111i·3-s + (0.984 + 0.173i)4-s + 1.33i·5-s + (0.00965 − 0.110i)6-s + 0.377·7-s + (−0.966 − 0.257i)8-s + 0.987·9-s + (0.115 − 1.32i)10-s + 0.302i·11-s + (−0.0192 + 0.109i)12-s + 0.277i·13-s + (−0.376 − 0.0328i)14-s − 0.148·15-s + (0.940 + 0.340i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.257 - 0.966i$
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: \((2,\ 728,\ (\ :1/2),\ 0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868483 + 0.667087i\)
\(L(\frac12)\) \(\approx\) \(0.868483 + 0.667087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.122i)T \)
7 \( 1 - T \)
13 \( 1 - iT \)
good3 \( 1 - 0.192iT - 3T^{2} \)
5 \( 1 - 2.98iT - 5T^{2} \)
11 \( 1 - 1.00iT - 11T^{2} \)
17 \( 1 - 5.01T + 17T^{2} \)
19 \( 1 + 3.05iT - 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 - 7.00iT - 29T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 + 5.29iT - 37T^{2} \)
41 \( 1 + 2.79T + 41T^{2} \)
43 \( 1 - 1.27iT - 43T^{2} \)
47 \( 1 + 3.77T + 47T^{2} \)
53 \( 1 - 3.46iT - 53T^{2} \)
59 \( 1 - 14.0iT - 59T^{2} \)
61 \( 1 - 6.82iT - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 7.60T + 71T^{2} \)
73 \( 1 - 9.74T + 73T^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 - 4.59iT - 83T^{2} \)
89 \( 1 - 1.30T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−10.65627557145270522551074378861, −9.771528271337828858499253374134, −9.060519281926982972751068134121, −7.81182463296602682588484583901, −7.17320112460257315235836634774, −6.64832396157558539851803819699, −5.33698213410301098954121249694, −3.79011774080424536365603791165, −2.76319395087627903803517147381, −1.50370640547492253154923536983, 0.881104589508133649517125513225, 1.82233798120479565534716230783, 3.59599481995497758488463943475, 4.94434932797933462137862198035, 5.77987133892329820490398685753, 6.94675379828268884313511003564, 8.070044004401310697437544992225, 8.227502858214908502497596490108, 9.503252003444826057232924507457, 9.875150882786280165269037838230

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