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Properties

Label 2-728-104.101-c1-0-19
Degree $2$
Conductor $728$
Sign $-0.129 - 0.991i$
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  + (−1.10 + 0.883i)2-s + (0.460 − 0.266i)3-s + (0.439 − 1.95i)4-s − 0.503·5-s + (−0.273 + 0.700i)6-s + (−0.866 − 0.5i)7-s + (1.23 + 2.54i)8-s + (−1.35 + 2.35i)9-s + (0.555 − 0.444i)10-s + (−0.0273 − 0.0472i)11-s + (−0.316 − 1.01i)12-s + (3.43 + 1.10i)13-s + (1.39 − 0.212i)14-s + (−0.231 + 0.133i)15-s + (−3.61 − 1.71i)16-s + (1.27 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.780 + 0.624i)2-s + (0.266 − 0.153i)3-s + (0.219 − 0.975i)4-s − 0.225·5-s + (−0.111 + 0.286i)6-s + (−0.327 − 0.188i)7-s + (0.437 + 0.899i)8-s + (−0.452 + 0.784i)9-s + (0.175 − 0.140i)10-s + (−0.00823 − 0.0142i)11-s + (−0.0914 − 0.293i)12-s + (0.952 + 0.305i)13-s + (0.373 − 0.0568i)14-s + (−0.0598 + 0.0345i)15-s + (−0.903 − 0.428i)16-s + (0.309 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576063 + 0.656321i\)
\(L(\frac12)\) \(\approx\) \(0.576063 + 0.656321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.883i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.43 - 1.10i)T \)
good3 \( 1 + (-0.460 + 0.266i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 0.503T + 5T^{2} \)
11 \( 1 + (0.0273 + 0.0472i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.27 + 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 - 4.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.685 - 1.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.07 + 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.99iT - 31T^{2} \)
37 \( 1 + (-4.90 - 8.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.92 - 1.68i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.84 - 2.79i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.59iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + (0.303 - 0.525i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.79 - 5.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0600 + 0.104i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.29 - 1.90i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.23iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 8.30T + 83T^{2} \)
89 \( 1 + (-0.726 + 0.419i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.37 + 3.67i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.39553424027774067090435258090, −9.761936204182185188146405276467, −8.561730704119471738041837212617, −8.239814606990164329113198249124, −7.28341230173790499063225228150, −6.36094926377583072546492053119, −5.53588112679414980003846725779, −4.31790440082089646335857924943, −2.82961213432034409346798388255, −1.38529069229249936173327593718, 0.60865239358444646106931277484, 2.35940721382442886311739220265, 3.41695900620524374705417242139, 4.20130902883395604033293177792, 5.91369364534143714689338301324, 6.74695826875443886939867821915, 7.922716050154404952526753037582, 8.621477475785884445193440845690, 9.268576955094180881798485861934, 10.08154258711273793060447564901

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