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Properties

Label 1-42e2-1764.1087-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.830 + 0.557i$
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  + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)11-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 − 0.149i)29-s + (0.5 + 0.866i)31-s + (−0.988 − 0.149i)37-s + (0.955 + 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.826 + 0.563i)47-s + (−0.988 + 0.149i)53-s + (0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)5-s + (0.900 + 0.433i)11-s + (0.826 − 0.563i)13-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 − 0.149i)29-s + (0.5 + 0.866i)31-s + (−0.988 − 0.149i)37-s + (0.955 + 0.294i)41-s + (−0.955 + 0.294i)43-s + (−0.826 + 0.563i)47-s + (−0.988 + 0.149i)53-s + (0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08133542435 + 0.2670564962i\)
\(L(\frac12)\) \(\approx\) \(0.08133542435 + 0.2670564962i\)
\(L(1)\) \(\approx\) \(0.9583593473 - 0.08677574269i\)
\(L(1)\) \(\approx\) \(0.9583593473 - 0.08677574269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (-0.955 + 0.294i)T \)
61 \( 1 + (0.365 + 0.930i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−19.698647254423662796456803892669, −18.888436047817813850329231074440, −18.46477142229225637127962270545, −17.61805344225343534520980332155, −16.83262179008872897676794875139, −15.903355137321213752109631455695, −15.44767440882359535247709634458, −14.37926210891954205800567047742, −13.95443100093290475752915600859, −13.271253836989668837345727238948, −11.97557408719494288808878519466, −11.421879123883952959228531159469, −10.99961817085366328552925616497, −9.78942276024420142015597601224, −9.30503216614721914192039190234, −8.27958953960056406132733119339, −7.39451977559657128317820781753, −6.70547834545815679931378512796, −6.02568884935934531811130752252, −5.020988640165683895528209800768, −3.80630349717016961476449848328, −3.40504099827913113748864675078, −2.3074586728641974549350379094, −1.26685447138014110705526495867, −0.050656117735597804895440158032, 1.249096170297150740961959962830, 1.69017960831403422049944879992, 3.27348822893476616874066065283, 3.96083041008882231499739589568, 4.74986313159244249340948196946, 5.76814410097366660105281208296, 6.33562695596895876986964671699, 7.561325525067738299757931816571, 8.24183449378708207543022690876, 8.88658177581130538729477624005, 9.75385629924176653626058051234, 10.50459680626380049295122666965, 11.497636096221748040666081911373, 12.286889705046141754613720983448, 12.709696156086483414595946985380, 13.62375443477031405539851296908, 14.46198629539510909755213525354, 15.17647931106221553914247749376, 16.13115762725151210010430774458, 16.539824070222179383672868442902, 17.41514047353500415664444022761, 18.01038569777847754721607972605, 19.0456596992604469471445363330, 19.65492729689495856419379799913, 20.48947560576103944217245222253

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