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Properties

Label 2-728-728.461-c1-0-104
Degree $2$
Conductor $728$
Sign $-0.946 - 0.322i$
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  + (−0.366 − 1.36i)2-s + (1.72 − 2.98i)3-s + (−1.73 + i)4-s + (1.25 − 1.25i)5-s + (−4.70 − 1.25i)6-s + (0.684 − 2.55i)7-s + (2 + 1.99i)8-s + (−4.42 − 7.65i)9-s + (−2.18 − 1.25i)10-s + 6.88i·12-s + (0.833 + 3.50i)13-s − 3.74·14-s + (−1.58 − 5.92i)15-s + (1.99 − 3.46i)16-s + (−8.84 + 8.84i)18-s + (2.32 + 0.623i)19-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)2-s + (0.993 − 1.72i)3-s + (−0.866 + 0.5i)4-s + (0.563 − 0.563i)5-s + (−1.91 − 0.514i)6-s + (0.258 − 0.965i)7-s + (0.707 + 0.707i)8-s + (−1.47 − 2.55i)9-s + (−0.689 − 0.398i)10-s + 1.98i·12-s + (0.231 + 0.972i)13-s − 0.999·14-s + (−0.409 − 1.52i)15-s + (0.499 − 0.866i)16-s + (−2.08 + 2.08i)18-s + (0.534 + 0.143i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.946 - 0.322i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309803 + 1.86722i\)
\(L(\frac12)\) \(\approx\) \(0.309803 + 1.86722i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
7 \( 1 + (-0.684 + 2.55i)T \)
13 \( 1 + (-0.833 - 3.50i)T \)
good3 \( 1 + (-1.72 + 2.98i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.25 + 1.25i)T - 5iT^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 - 0.623i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-8.21 - 4.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (3.79 - 14.1i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.28 + 2.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-10.5 - 2.82i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + (11.4 - 11.4i)T - 83iT^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-84.0 - 48.5i)T^{2} \)
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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

−9.525062613392203510268600356425, −9.163469260019598053741562196637, −8.285763463637249931235075809945, −7.45648117074566290047986873381, −6.81088217402330940766280690218, −5.37031430030668682052148240410, −3.91146410822090346226127469986, −2.90339413237673656285114530742, −1.63206469494266028296720689634, −1.09888698547469099368664323669, 2.51707975235063092831241712790, 3.43473318335613041380828289566, 4.81754025380836867521252305799, 5.29144798429472872019036135710, 6.32126300364868165641156416650, 7.74379636958466281343933140785, 8.494746636576093791017231974787, 9.100870315505772204056549483031, 9.771333070715960099247297730317, 10.51850109171955211127460187657

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