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Properties

Label 1-648-648.547-r1-0-0
Degree $1$
Conductor $648$
Sign $0.713 - 0.700i$
Analytic cond. $69.6372$
Root an. cond. $69.6372$
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  + (−0.597 − 0.802i)5-s + (0.835 − 0.549i)7-s + (−0.993 − 0.116i)11-s + (0.286 + 0.957i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.835 + 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.686 + 0.727i)29-s + (0.0581 − 0.998i)31-s + (−0.939 − 0.342i)35-s + (0.939 − 0.342i)37-s + (0.973 + 0.230i)41-s + (0.396 + 0.918i)43-s + (0.0581 + 0.998i)47-s + ⋯
L(s)  = 1  + (−0.597 − 0.802i)5-s + (0.835 − 0.549i)7-s + (−0.993 − 0.116i)11-s + (0.286 + 0.957i)13-s + (0.173 − 0.984i)17-s + (0.173 + 0.984i)19-s + (0.835 + 0.549i)23-s + (−0.286 + 0.957i)25-s + (0.686 + 0.727i)29-s + (0.0581 − 0.998i)31-s + (−0.939 − 0.342i)35-s + (0.939 − 0.342i)37-s + (0.973 + 0.230i)41-s + (0.396 + 0.918i)43-s + (0.0581 + 0.998i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ 0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.784995327 - 0.7292508374i\)
\(L(\frac12)\) \(\approx\) \(1.784995327 - 0.7292508374i\)
\(L(1)\) \(\approx\) \(1.072088190 - 0.1926570766i\)
\(L(1)\) \(\approx\) \(1.072088190 - 0.1926570766i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.835 - 0.549i)T \)
11 \( 1 + (-0.993 - 0.116i)T \)
13 \( 1 + (0.286 + 0.957i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.835 + 0.549i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
37 \( 1 + (0.939 - 0.342i)T \)
41 \( 1 + (0.973 + 0.230i)T \)
43 \( 1 + (0.396 + 0.918i)T \)
47 \( 1 + (0.0581 + 0.998i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.993 + 0.116i)T \)
61 \( 1 + (-0.893 - 0.448i)T \)
67 \( 1 + (-0.686 + 0.727i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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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

−22.93400228199932244770901243855, −21.837164721739510589062809789999, −21.27662514005296228905384912365, −20.29093583240262377704363569064, −19.44002941859099624824490759613, −18.52643891763154372313262647192, −17.99654267563767727502422022555, −17.18929639129816016689927018575, −15.77498012545052371830150718868, −15.311868849948612591767171675156, −14.68273153711578233025137255492, −13.59622027328824111788020823125, −12.6288134597569616824957849039, −11.78246278860105245158562012157, −10.69812833836178728879694782466, −10.51473473786382131032803870800, −8.92134931069373889212400254165, −8.07793081594177066039920021885, −7.46243773519788121242744121175, −6.282523375970184075465182330066, −5.29572651013341240067189286028, −4.35902159212383179125063231057, −3.05613899703440188128395750422, −2.38783845537661154395034282477, −0.797405599294145968410422462514, 0.664265095706620630721174184757, 1.620129587699894127762640051960, 3.05878545985422142010407991401, 4.30233496508771085971703511469, 4.85366985798193789167036391956, 5.907351171861841132124844081001, 7.42011791211351220678558937235, 7.80530558163987051537979152177, 8.83629930988619320954867129540, 9.71623144571029677348313910861, 10.96944353238150702083130991157, 11.49652606086512963777785075667, 12.47218829006521127654849498838, 13.38378310894061542409968394758, 14.16654590331867066245078375782, 15.10402986009052209829776389759, 16.24391857373084977178024417088, 16.47710594479935315138660020655, 17.645486633029935022795853062986, 18.45766323971552304165917101092, 19.30337013314469127675616729263, 20.28668613837558962127962144583, 20.94493931204930278390477185516, 21.36262988612950879614668078380, 22.92061256785040204523829406958

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