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Properties

Label 1-42e2-1764.1139-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.236 + 0.971i$
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  + (0.0747 − 0.997i)5-s + (0.365 − 0.930i)11-s + (−0.365 + 0.930i)13-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.955 − 0.294i)29-s + 31-s + (−0.733 + 0.680i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.623 − 0.781i)47-s + (0.733 + 0.680i)53-s + (−0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)5-s + (0.365 − 0.930i)11-s + (−0.365 + 0.930i)13-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.955 − 0.294i)29-s + 31-s + (−0.733 + 0.680i)37-s + (0.826 + 0.563i)41-s + (−0.826 + 0.563i)43-s + (−0.623 − 0.781i)47-s + (0.733 + 0.680i)53-s + (−0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.236 + 0.971i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.236 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2511144391 + 0.3195308671i\)
\(L(\frac12)\) \(\approx\) \(0.2511144391 + 0.3195308671i\)
\(L(1)\) \(\approx\) \(0.9140579864 - 0.1858863496i\)
\(L(1)\) \(\approx\) \(0.9140579864 - 0.1858863496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (-0.365 + 0.930i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (0.733 + 0.680i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.365 - 0.930i)T \)
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

−19.73455930957260076968212005502, −19.1219735018730889976470911446, −18.340622169441621530101201157804, −17.65724860874022936488994590877, −17.110873420237541626500617486972, −16.046082996831712380747706187236, −15.28276412639509401937225074602, −14.60304501741205389100519782287, −14.158560125795851598160356343639, −13.09796503877754710682446749614, −12.28465155117752731604945110272, −11.681425240702826388309492956796, −10.65229669010730223250888091428, −10.039039839561552995732695686725, −9.53380056727401336847722606786, −8.21734515189036798910872548588, −7.54417301182560948783523341871, −6.90259694356204594817191901964, −5.91890519100405357913302012158, −5.263661610987685564468136974037, −4.01564284529608905189060370326, −3.361224553546578733592481915339, −2.36951786630328508865568311719, −1.53634662763151242949185809930, −0.07849800686132811189633791644, 0.938611312548517887085421450952, 1.81288423215910225318977162217, 2.91437749772455623180084816691, 4.04328954198418080938104869051, 4.62172389092704787324438809841, 5.63503978828310315896117444441, 6.296391647213138986853302305047, 7.2909428866029208487393228657, 8.35355921198812957339228345576, 8.74174527585095159440576342773, 9.64720133818384105880416039527, 10.35284094057786355946043180355, 11.61882305023071162745712156301, 11.83425010742678328618971178280, 12.89897509105120210647797009040, 13.50708786326146632352675597390, 14.27780937267883540113222036301, 15.0547248664971912194997595432, 16.07838635940212893526798493442, 16.658347792159138054372565536119, 17.05782918109603731726340523812, 18.028952654778496195008144814328, 19.02461367750526494928032713838, 19.444640677307359374721151806129, 20.25637303945802759313201674200

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