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Properties

Label 1-42e2-1764.191-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.763 - 0.645i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.826 − 0.563i)29-s + (0.5 + 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.733 − 0.680i)47-s + (−0.826 + 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)5-s + (−0.222 + 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.0747 + 0.997i)17-s + (0.5 + 0.866i)19-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.826 − 0.563i)29-s + (0.5 + 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.733 − 0.680i)47-s + (−0.826 + 0.563i)53-s + (−0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.763 - 0.645i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03239672784 + 0.08851844984i\)
\(L(\frac12)\) \(\approx\) \(-0.03239672784 + 0.08851844984i\)
\(L(1)\) \(\approx\) \(0.7133163631 + 0.1739110482i\)
\(L(1)\) \(\approx\) \(0.7133163631 + 0.1739110482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
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

−19.827002252664090522446860232374, −19.15054155717522396976344013678, −18.39785456132904990033071057533, −17.556274372961191788578946056508, −16.549733888988779190013648165602, −16.29144393383362747861177330356, −15.48336718878163466138874544620, −14.62586521651786104059984304313, −13.67279186260745670610890143031, −13.211023816712670396834091452908, −12.15737314998597363171962545827, −11.56526151148089325510138597640, −11.045263243116580693760083688647, −9.664606531792611377476948923024, −9.280365605998328671505682053496, −8.31553481380852191957096868066, −7.63665032410273225996175598965, −6.85945378927849821090766118059, −5.7217933936892196817746261415, −4.993018536205510503332207995329, −4.23821357808988996965388086797, −3.29864662920457801821911118291, −2.34139282356360397750871333024, −1.10238638668019998698951418542, −0.03588707777845026991880061205, 1.65425102624786700742801039087, 2.55278853959068870554600915600, 3.48002115014368673787797524945, 4.25090468752340168881146434067, 5.1839006761873556121788504130, 6.20091318831085473825586419961, 6.9414797232963392625540633393, 7.920944914318992540839057895517, 8.11762519326373576930472008810, 9.64309289548313137337083193495, 10.14550592770909347865490646264, 10.82504723981815731590375734977, 11.84571922804367455503691874666, 12.36008154161340683672583597162, 13.13710276501475770486439287961, 14.28851547765893882373395733536, 14.7825515130309439517456574231, 15.40976751752010469544134748938, 16.13022379760772243876345166452, 17.14402708202226366391220442446, 17.78726243315465488007681473149, 18.51481859200991356610985033853, 19.19015356688040788043175920891, 20.04291907994820049091910996175, 20.42269117736521122901633076377

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