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Properties

Label 1-42e2-1764.799-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.0676 - 0.997i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.955 + 0.294i)5-s + (−0.0747 − 0.997i)11-s + (0.826 − 0.563i)13-s + (0.623 − 0.781i)17-s − 19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.988 − 0.149i)29-s + (0.5 − 0.866i)31-s + (0.623 − 0.781i)37-s + (0.955 + 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.0747 − 0.997i)47-s + (0.623 + 0.781i)53-s + (0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)5-s + (−0.0747 − 0.997i)11-s + (0.826 − 0.563i)13-s + (0.623 − 0.781i)17-s − 19-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.988 − 0.149i)29-s + (0.5 − 0.866i)31-s + (0.623 − 0.781i)37-s + (0.955 + 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.0747 − 0.997i)47-s + (0.623 + 0.781i)53-s + (0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0676 - 0.997i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.0676 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.738490132 - 1.860312990i\)
\(L(\frac12)\) \(\approx\) \(1.738490132 - 1.860312990i\)
\(L(1)\) \(\approx\) \(1.290916927 - 0.2471304164i\)
\(L(1)\) \(\approx\) \(1.290916927 - 0.2471304164i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
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

−20.48678063236770654849892733258, −19.44077471443828847559700163927, −18.76229566847266939118602920277, −17.96443554009803429192448531831, −17.24903967266224629892959447802, −16.76177729613374751027092417825, −15.87556817356637784093369415132, −14.868207247578507290213645041156, −14.45900069692263220293610156489, −13.33926767227976019069499968054, −12.95615426952599009858385544070, −12.17596724554093005725431533031, −11.12607955662288675199730677824, −10.38659670520484343010544894222, −9.6798490402325867651525774309, −8.91482249685203514277044289432, −8.225209203866725156434479876346, −7.07635040569100507499920969560, −6.39465350413672965340153362404, −5.60237265573700641820443509579, −4.7366139923624318504073025497, −3.92449431851462590811641637667, −2.766786487050267734132073762793, −1.77856634326414012539681572704, −1.19517966305707746014556626252, 0.45066636041282437455656822390, 1.35598916472906699522516598750, 2.52360497458594724186487734040, 3.16242809736061024701856615193, 4.201411782828767813235103268052, 5.42528902977698245004545110729, 5.86640104942313685082087551258, 6.66614303645066731964995047193, 7.64604058358032572943316301473, 8.55785927717361729787147311039, 9.24646878816389933385412568397, 10.07442960377202382734518647171, 10.908016765902765008441354126949, 11.33720607660142532773054027654, 12.586378546165219719795566597497, 13.336727713238715174924887716237, 13.726372592028137928024770093924, 14.71860536651946289357987436678, 15.2603759646891786141852999577, 16.47140404465880690766560091169, 16.76350350680196195991567550786, 17.73513092781812781427507559183, 18.49927481420885007418938661662, 18.861963240050503655957740861692, 19.88846439020109539651325335646

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