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Properties

Label 1-42e2-1764.1327-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.805 - 0.592i$
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  + (−0.900 − 0.433i)5-s + (−0.623 + 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.955 − 0.294i)17-s + (0.5 − 0.866i)19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 − 0.680i)29-s + (0.5 − 0.866i)31-s + (−0.733 − 0.680i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.988 + 0.149i)47-s + (−0.733 + 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)5-s + (−0.623 + 0.781i)11-s + (−0.988 − 0.149i)13-s + (0.955 − 0.294i)17-s + (0.5 − 0.866i)19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 − 0.680i)29-s + (0.5 − 0.866i)31-s + (−0.733 − 0.680i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (0.988 + 0.149i)47-s + (−0.733 + 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1540010044 - 0.4694108237i\)
\(L(\frac12)\) \(\approx\) \(0.1540010044 - 0.4694108237i\)
\(L(1)\) \(\approx\) \(0.7847697539 - 0.06073456429i\)
\(L(1)\) \(\approx\) \(0.7847697539 - 0.06073456429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (-0.0747 + 0.997i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
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

−20.45323003873664039735915247627, −19.39301774352450348109366174924, −18.89431666375262165241895956162, −18.45354522271752771262711869571, −17.31949252277139124579583846814, −16.55116542761027651431780123256, −15.97498451188676780236915042085, −15.160651495823920294555268721, −14.39189210687643360314118755212, −13.916152623061133364770855517531, −12.57670483906817611311342070438, −12.25001534894053233289187759277, −11.361427581528523397236113286754, −10.50982293664282183388541048887, −10.0372676976158913391737343050, −8.78068008377271412431082584381, −8.1262110984866164343866266784, −7.39820683000980879891466929145, −6.696181779350803921309040889485, −5.56371662806577704344238243312, −4.905106005282664123595810246678, −3.71700393829024462989618056502, −3.20221781079475244778605953797, −2.19459483522384253192294868715, −0.84660966458187308165514368905, 0.12177938206262494881041840459, 1.092088686710359331922184772, 2.368036507458980950456345922279, 3.20359701257526524068258902850, 4.22917993354415279624989534714, 4.9713777713548419759398283280, 5.6066130088719695383628914251, 7.00133179750832618781310270664, 7.61581510021839160853281683297, 8.045544969562519001020350112441, 9.402682111291090413375966503607, 9.67399781436933841561189074443, 10.84495247956749530365873155829, 11.623511286565923046505904148707, 12.25739273926868118798222280628, 12.92261429160329650169192528017, 13.74040918658394612410310843344, 14.81677179143181072983030110051, 15.32810973401518642272268258558, 15.96075159466240184935380700882, 16.860564711984363419174077610502, 17.45933811204245286919880653685, 18.33987958961604423064993540199, 19.18765348270499174210523741587, 19.71838014386502539388009254594

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