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Properties

Label 1-72e2-5184.403-r1-0-0
Degree $1$
Conductor $5184$
Sign $0.965 - 0.259i$
Analytic cond. $557.098$
Root an. cond. $557.098$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.653 + 0.756i)5-s + (0.474 + 0.880i)7-s + (−0.900 + 0.435i)11-s + (−0.244 − 0.969i)13-s + (−0.642 + 0.766i)17-s + (−0.953 + 0.300i)19-s + (−0.474 + 0.880i)23-s + (−0.144 − 0.989i)25-s + (0.999 − 0.0145i)29-s + (−0.686 + 0.727i)31-s + (−0.976 − 0.216i)35-s + (−0.216 − 0.976i)37-s + (−0.784 − 0.620i)41-s + (−0.827 − 0.561i)43-s + (−0.727 + 0.686i)47-s + ⋯
L(s)  = 1  + (−0.653 + 0.756i)5-s + (0.474 + 0.880i)7-s + (−0.900 + 0.435i)11-s + (−0.244 − 0.969i)13-s + (−0.642 + 0.766i)17-s + (−0.953 + 0.300i)19-s + (−0.474 + 0.880i)23-s + (−0.144 − 0.989i)25-s + (0.999 − 0.0145i)29-s + (−0.686 + 0.727i)31-s + (−0.976 − 0.216i)35-s + (−0.216 − 0.976i)37-s + (−0.784 − 0.620i)41-s + (−0.827 − 0.561i)43-s + (−0.727 + 0.686i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(557.098\)
Root analytic conductor: \(557.098\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (1:\ ),\ 0.965 - 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06285236312 + 0.008313423471i\)
\(L(\frac12)\) \(\approx\) \(0.06285236312 + 0.008313423471i\)
\(L(1)\) \(\approx\) \(0.6366466711 + 0.2419946912i\)
\(L(1)\) \(\approx\) \(0.6366466711 + 0.2419946912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.653 + 0.756i)T \)
7 \( 1 + (0.474 + 0.880i)T \)
11 \( 1 + (-0.900 + 0.435i)T \)
13 \( 1 + (-0.244 - 0.969i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (-0.953 + 0.300i)T \)
23 \( 1 + (-0.474 + 0.880i)T \)
29 \( 1 + (0.999 - 0.0145i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
37 \( 1 + (-0.216 - 0.976i)T \)
41 \( 1 + (-0.784 - 0.620i)T \)
43 \( 1 + (-0.827 - 0.561i)T \)
47 \( 1 + (-0.727 + 0.686i)T \)
53 \( 1 + (-0.991 - 0.130i)T \)
59 \( 1 + (0.328 + 0.944i)T \)
61 \( 1 + (-0.159 + 0.987i)T \)
67 \( 1 + (-0.696 - 0.717i)T \)
71 \( 1 + (-0.422 + 0.906i)T \)
73 \( 1 + (0.422 + 0.906i)T \)
79 \( 1 + (0.116 - 0.993i)T \)
83 \( 1 + (0.873 + 0.487i)T \)
89 \( 1 + (-0.906 + 0.422i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.825921870861533630391094814113, −16.99949277111244910248373854706, −16.51667574441175131039962283305, −16.03545478612716915238166117485, −15.224769399518506865539800172733, −14.55986630479734174143785138398, −13.68338828421510711788257018618, −13.30094206464304540780288337194, −12.540675279290275347098823388428, −11.69028066123776209538657236598, −11.254224022664018608408991648760, −10.536268197163438727863991016183, −9.766165066890044298825311331432, −8.92254638518062300668341870425, −8.202934751270686808549384209752, −7.87187087862927129068919478545, −6.84436011073936389413859126052, −6.41088968169826867839681864653, −4.97945925064644729316427155806, −4.79459738967080172607941604010, −4.10353751328951110030288659664, −3.21350094121228958710328211063, −2.21662743191579997961649048871, −1.410740374183125868027853931273, −0.31165713889609393329631490603, 0.022430060711742302802560157, 1.58911242030343446624444936273, 2.31385220345682104386048978159, 2.98132415129394396377095392675, 3.79562979558997988190657932178, 4.64710108430786560505151889687, 5.40047832724673603897190045148, 6.0637980319978994682896064701, 6.92487873005559635180479146810, 7.65569366030986971324664228008, 8.26784340741793836047020992452, 8.735262369771279217792388736050, 9.89168895628101288884277029886, 10.53947607810893499224237005042, 10.935918657901731243645977346787, 11.83194360535329345917382129661, 12.43587523518831076089429610256, 12.94128034895557388805888732436, 13.91110971048758888417102506932, 14.713637039117519508052615582496, 15.21667865306421305112690357069, 15.55667915335446421699308816100, 16.25484515757046699742620586542, 17.43875948029996309662338840472, 17.87604271514912086446983760482

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