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Properties

Label 2-42e2-21.11-c2-0-8
Degree $2$
Conductor $1764$
Sign $0.300 - 0.953i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.46 − 2.57i)5-s + (−5.25 − 3.03i)11-s − 12.5·13-s + (−14.9 − 8.64i)17-s + (12.9 + 22.4i)19-s + (−2.09 + 1.20i)23-s + (0.791 − 1.37i)25-s + 55.8i·29-s + (−7.64 + 13.2i)31-s + (11.8 + 20.5i)37-s + 15.4i·41-s + 27.7·43-s + (24.6 − 14.2i)47-s + (40.4 + 23.3i)53-s − 31.2·55-s + ⋯
L(s)  = 1  + (0.893 − 0.515i)5-s + (−0.477 − 0.275i)11-s − 0.967·13-s + (−0.881 − 0.508i)17-s + (0.680 + 1.17i)19-s + (−0.0909 + 0.0525i)23-s + (0.0316 − 0.0548i)25-s + 1.92i·29-s + (−0.246 + 0.427i)31-s + (0.320 + 0.555i)37-s + 0.377i·41-s + 0.645·43-s + (0.524 − 0.302i)47-s + (0.762 + 0.440i)53-s − 0.568·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.300 - 0.953i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.577584014\)
\(L(\frac12)\) \(\approx\) \(1.577584014\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-4.46 + 2.57i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (5.25 + 3.03i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 12.5T + 169T^{2} \)
17 \( 1 + (14.9 + 8.64i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-12.9 - 22.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (2.09 - 1.20i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 55.8iT - 841T^{2} \)
31 \( 1 + (7.64 - 13.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.8 - 20.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 15.4iT - 1.68e3T^{2} \)
43 \( 1 - 27.7T + 1.84e3T^{2} \)
47 \( 1 + (-24.6 + 14.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-40.4 - 23.3i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-69.8 - 40.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-31.3 - 54.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (58.0 - 100. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 49.7iT - 5.04e3T^{2} \)
73 \( 1 + (15.3 - 26.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (71.3 + 123. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 28.5iT - 6.88e3T^{2} \)
89 \( 1 + (-145. + 84.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 113.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.166137761153686302930992036884, −8.728340245327749987985794249329, −7.60165924463382331363380633342, −6.98612881852569520886452595831, −5.82570809632933033344853962578, −5.32162988857529341160763664227, −4.49038741722839094133047330702, −3.21094266030583486616853868151, −2.22604554468090849327252772418, −1.16532087642535368744021798228, 0.41588606050581486574746672629, 2.23870632122980220889183087931, 2.50183726902091956643184978653, 3.98531107234605466333849998628, 4.91970799743965439449150806480, 5.76447807987978255204504988328, 6.55625488493050735250358279056, 7.31149975755821059466116860168, 8.081048331740358749759549394281, 9.200638922804296236917128109300

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