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Properties

Label 2-728-13.4-c1-0-3
Degree $2$
Conductor $728$
Sign $-0.588 - 0.808i$
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  + (−0.266 + 0.461i)3-s + 3.61i·5-s + (0.866 − 0.5i)7-s + (1.35 + 2.35i)9-s + (0.788 + 0.455i)11-s + (−2.94 + 2.08i)13-s + (−1.66 − 0.961i)15-s + (−1.46 − 2.54i)17-s + (−0.223 + 0.128i)19-s + 0.532i·21-s + (2.40 − 4.16i)23-s − 8.03·25-s − 3.04·27-s + (−1.09 + 1.90i)29-s + 7.13i·31-s + ⋯
L(s)  = 1  + (−0.153 + 0.266i)3-s + 1.61i·5-s + (0.327 − 0.188i)7-s + (0.452 + 0.784i)9-s + (0.237 + 0.137i)11-s + (−0.815 + 0.578i)13-s + (−0.430 − 0.248i)15-s + (−0.356 − 0.616i)17-s + (−0.0511 + 0.0295i)19-s + 0.116i·21-s + (0.501 − 0.869i)23-s − 1.60·25-s − 0.586·27-s + (−0.203 + 0.353i)29-s + 1.28i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584104 + 1.14785i\)
\(L(\frac12)\) \(\approx\) \(0.584104 + 1.14785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.94 - 2.08i)T \)
good3 \( 1 + (0.266 - 0.461i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.61iT - 5T^{2} \)
11 \( 1 + (-0.788 - 0.455i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.46 + 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.223 - 0.128i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.40 + 4.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.09 - 1.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.13iT - 31T^{2} \)
37 \( 1 + (-3.59 - 2.07i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.85 + 3.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.906 - 1.57i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.764iT - 47T^{2} \)
53 \( 1 + 7.04T + 53T^{2} \)
59 \( 1 + (6.34 - 3.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.99 - 8.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.3 - 7.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.36 + 4.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.78iT - 73T^{2} \)
79 \( 1 - 4.79T + 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 + (-10.0 - 5.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.96 + 2.28i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.64065162291751471333214566404, −10.11925724954578387739473690522, −9.147341022105353236022494848776, −7.88844986807242239272383619223, −7.03639689844548068455035409745, −6.63402593610842854509785188142, −5.17426900211096061161191987986, −4.33107106387527746461030994294, −3.04792404172674938159857337914, −2.03916678626320606262139182855, 0.68392144927169100788031512065, 1.89390768909033850074157485779, 3.70526919988002205977831360365, 4.70922676452823944282425513068, 5.49374548185702454583761383475, 6.47309958953563714620231405666, 7.69853751380023510684337968112, 8.333888916663108048514502707557, 9.378452936398161017743897015164, 9.706376871710185809226462285997

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