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Properties

Label 1-2736-2736.835-r0-0-0
Degree $1$
Conductor $2736$
Sign $0.737 + 0.675i$
Analytic cond. $12.7059$
Root an. cond. $12.7059$
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  + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)13-s + 17-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s − i·35-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)13-s + 17-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.5 + 0.866i)31-s − i·35-s − i·37-s + (−0.5 + 0.866i)41-s + (0.866 − 0.5i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(12.7059\)
Root analytic conductor: \(12.7059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2736,\ (0:\ ),\ 0.737 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.949228995 + 0.7580136998i\)
\(L(\frac12)\) \(\approx\) \(1.949228995 + 0.7580136998i\)
\(L(1)\) \(\approx\) \(1.301681890 + 0.1403039488i\)
\(L(1)\) \(\approx\) \(1.301681890 + 0.1403039488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.0564281592582740433246688660, −18.464071147421498245062167650039, −17.778971815631373394638591935228, −17.07238202174590774918446116339, −16.3552562801470293980113467438, −15.80589690731089469785045680870, −14.751311582822044699480333848420, −14.385856772561178368051436943538, −13.29348491683412266693828727410, −12.84743241372416884905824529480, −12.141481126452944949192140061795, −11.45919766476454835386278233964, −10.30941118955868387501105773923, −9.81440376585789147673868872601, −9.00871444715817034131820719639, −8.58548099194539439020346223980, −7.52623766584111512544477596322, −6.54893453960145172333216182578, −5.75158620057516789366584496888, −5.53012276960829494543241582216, −4.25914726867826804207506535837, −3.52779126963383553703788518135, −2.41242405329261657581857860770, −1.78907845911464351274298212594, −0.70160404030634384622569964510, 1.18069016981628548064100637553, 1.62670525866884197424816970958, 3.07155630329305328724005310882, 3.51327248108672293744523715982, 4.3769202581260118194789768265, 5.65011459230215714091546862167, 6.08233611241399738830754381059, 6.91173578437315164938409968302, 7.486220682362455550399850737898, 8.637537357466361655211746327651, 9.35681368748415656221431544467, 9.971613172980717132188966599617, 10.71389323477103707533548468710, 11.33661624434321036710106407094, 12.21264460569226113780003508915, 13.20144956864395819043379198671, 13.72166602711913712214377812942, 14.23943156047968545861536479145, 14.90157991801383304859261410616, 16.091812423643159446576464639599, 16.51716573461693034740684854621, 17.23460065068821108626326714699, 17.84576602706538194915475168484, 18.83749144503810161791330606647, 19.09469300752701706024225214657

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