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Properties

Label 2-42e2-21.2-c2-0-6
Degree $2$
Conductor $1764$
Sign $0.999 + 0.0348i$
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  + (−4.89 − 2.82i)5-s + (1.22 − 0.707i)11-s − 16·13-s + (−14.6 + 8.48i)17-s + (−4 + 6.92i)19-s + (11.0 + 6.36i)23-s + (3.49 + 6.06i)25-s − 24.0i·29-s + (28 + 48.4i)31-s + (−12 + 20.7i)37-s − 50.9i·41-s + 40·43-s + (9.79 + 5.65i)47-s + (37.9 − 21.9i)53-s − 8·55-s + ⋯
L(s)  = 1  + (−0.979 − 0.565i)5-s + (0.111 − 0.0642i)11-s − 1.23·13-s + (−0.864 + 0.499i)17-s + (−0.210 + 0.364i)19-s + (0.479 + 0.276i)23-s + (0.139 + 0.242i)25-s − 0.829i·29-s + (0.903 + 1.56i)31-s + (−0.324 + 0.561i)37-s − 1.24i·41-s + 0.930·43-s + (0.208 + 0.120i)47-s + (0.716 − 0.413i)53-s − 0.145·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0348i$
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: \((2,\ 1764,\ (\ :1),\ 0.999 + 0.0348i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.108507590\)
\(L(\frac12)\) \(\approx\) \(1.108507590\)
\(L(2)\) 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 \( 1 + (4.89 + 2.82i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + (14.6 - 8.48i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-11.0 - 6.36i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 24.0iT - 841T^{2} \)
31 \( 1 + (-28 - 48.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (12 - 20.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 50.9iT - 1.68e3T^{2} \)
43 \( 1 - 40T + 1.84e3T^{2} \)
47 \( 1 + (-9.79 - 5.65i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-37.9 + 21.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (9.79 - 5.65i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20 - 34.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-13 - 22.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 134. iT - 5.04e3T^{2} \)
73 \( 1 + (44 + 76.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (41 - 71.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 101. iT - 6.88e3T^{2} \)
89 \( 1 + (53.8 + 31.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
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

−8.896008665478431253550358915519, −8.408864103525983140285572958004, −7.53861907147353873602427857932, −6.89899102938230091797500897188, −5.84906355342773708557797053735, −4.77516375120907958086609824480, −4.28552605306789125135211272141, −3.21722417026710628106044498867, −2.03439647130152091312983862794, −0.59617477347347157010533631715, 0.52213567004532524772303470214, 2.29013538117599211803478682538, 3.07165319959732803805232893240, 4.22523701002278110151681656286, 4.79212825007632200159464038723, 5.98113391507046125708665951548, 7.05127430460807152566254865097, 7.32718270989387138091844950624, 8.257642652110854330036280010605, 9.125074695944763134576306589559

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