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Properties

Label 2-42e2-21.2-c2-0-7
Degree $2$
Conductor $1764$
Sign $0.976 + 0.216i$
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  + (−3.67 − 2.12i)5-s + (−18.3 + 10.6i)11-s − 23·13-s + (14.6 − 8.48i)17-s + (−0.5 + 0.866i)19-s + (14.6 + 8.48i)23-s + (−3.5 − 6.06i)25-s + 33.9i·29-s + (−24.5 − 42.4i)31-s + (−8.5 + 14.7i)37-s + 21.2i·41-s + 47·43-s + (33.0 + 19.0i)47-s + (73.4 − 42.4i)53-s + 90.0·55-s + ⋯
L(s)  = 1  + (−0.734 − 0.424i)5-s + (−1.67 + 0.964i)11-s − 1.76·13-s + (0.864 − 0.499i)17-s + (−0.0263 + 0.0455i)19-s + (0.638 + 0.368i)23-s + (−0.140 − 0.242i)25-s + 1.17i·29-s + (−0.790 − 1.36i)31-s + (−0.229 + 0.397i)37-s + 0.517i·41-s + 1.09·43-s + (0.703 + 0.406i)47-s + (1.38 − 0.800i)53-s + 1.63·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.976 + 0.216i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9347916951\)
\(L(\frac12)\) \(\approx\) \(0.9347916951\)
\(L(2)\) 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 \( 1 + (3.67 + 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (18.3 - 10.6i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 23T + 169T^{2} \)
17 \( 1 + (-14.6 + 8.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.6 - 8.48i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9iT - 841T^{2} \)
31 \( 1 + (24.5 + 42.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (8.5 - 14.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 21.2iT - 1.68e3T^{2} \)
43 \( 1 - 47T + 1.84e3T^{2} \)
47 \( 1 + (-33.0 - 19.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-73.4 + 42.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-44.0 + 25.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20 - 34.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (11.5 + 19.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 63.6iT - 5.04e3T^{2} \)
73 \( 1 + (-8.5 - 14.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 106. iT - 6.88e3T^{2} \)
89 \( 1 + (117. + 67.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 40T + 9.40e3T^{2} \)
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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

−9.178947429322342764837496549893, −8.036589459966816649451706717664, −7.52601173152073941491735499472, −7.10455099916540475441405928553, −5.51257869066801052643785962390, −5.03293898483788960375020917772, −4.26188022874935682319483293601, −2.98185658966578036241604021296, −2.16757939574967693298461286552, −0.47975735589323419389726863931, 0.52539490950518679342194837405, 2.38266801698038916967671013634, 3.06518470381542959935851603200, 4.07590513822673801082334311025, 5.22876009306311208890975840421, 5.67279168924359081620295508691, 7.08029496324022946637661575880, 7.52709037571007111437318325814, 8.182287252028272107735830099770, 9.069368949741190585654442680722

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