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Properties

Label 2-728-13.10-c1-0-6
Degree $2$
Conductor $728$
Sign $-0.747 - 0.664i$
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  + (1.41 + 2.44i)3-s + 4.08i·5-s + (0.866 + 0.5i)7-s + (−2.48 + 4.30i)9-s + (−0.331 + 0.191i)11-s + (2.36 − 2.72i)13-s + (−10.0 + 5.77i)15-s + (2.13 − 3.68i)17-s + (5.59 + 3.23i)19-s + 2.82i·21-s + (−2.08 − 3.61i)23-s − 11.7·25-s − 5.58·27-s + (0.997 + 1.72i)29-s − 9.98i·31-s + ⋯
L(s)  = 1  + (0.815 + 1.41i)3-s + 1.82i·5-s + (0.327 + 0.188i)7-s + (−0.829 + 1.43i)9-s + (−0.0998 + 0.0576i)11-s + (0.655 − 0.755i)13-s + (−2.58 + 1.49i)15-s + (0.516 − 0.894i)17-s + (1.28 + 0.741i)19-s + 0.616i·21-s + (−0.435 − 0.753i)23-s − 2.34·25-s − 1.07·27-s + (0.185 + 0.320i)29-s − 1.79i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742646 + 1.95217i\)
\(L(\frac12)\) \(\approx\) \(0.742646 + 1.95217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-2.36 + 2.72i)T \)
good3 \( 1 + (-1.41 - 2.44i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 4.08iT - 5T^{2} \)
11 \( 1 + (0.331 - 0.191i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.13 + 3.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.59 - 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.08 + 3.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.997 - 1.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.98iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.13 - 1.80i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.59 + 7.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.162iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + (0.759 + 0.438i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.25 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.65 + 4.42i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.05 + 1.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.58iT - 73T^{2} \)
79 \( 1 - 8.34T + 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + (-1.75 + 1.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.91 - 2.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.57319563063104464878307326414, −9.935781667875396352687127741247, −9.314937465425089499191248248350, −8.041042301663574091372366627981, −7.55050233115144371140408176380, −6.21403058576298235551498023154, −5.27821863777105466008881630056, −3.94883030641366257358495736319, −3.22474287069478854245041322844, −2.50468007179128168757017228535, 1.14029033297699686536068982706, 1.66888053068835537877290097115, 3.33764148412050348613452235630, 4.59586282047660180756908267923, 5.61730790263087768320503773446, 6.68804165967450439398846320022, 7.80780102321159553833258073742, 8.200975523304545738880292330425, 8.991133322586975162677140858760, 9.619944993494529791217341132803

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