Properties

Label 2-42e2-21.2-c2-0-21
Degree $2$
Conductor $1764$
Sign $-0.807 + 0.590i$
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  + (3.08 + 1.77i)5-s + (−8.03 + 4.63i)11-s − 4.24·13-s + (2.11 − 1.22i)17-s + (−10.3 + 17.9i)19-s + (−0.682 − 0.394i)23-s + (−6.17 − 10.6i)25-s − 7.69i·29-s + (−11.7 − 20.4i)31-s + (11.6 − 20.2i)37-s − 51.5i·41-s − 49.3·43-s + (−13.6 − 7.88i)47-s + (−65.1 + 37.6i)53-s − 32.9·55-s + ⋯
L(s)  = 1  + (0.616 + 0.355i)5-s + (−0.730 + 0.421i)11-s − 0.326·13-s + (0.124 − 0.0718i)17-s + (−0.546 + 0.945i)19-s + (−0.0296 − 0.0171i)23-s + (−0.246 − 0.427i)25-s − 0.265i·29-s + (−0.380 − 0.658i)31-s + (0.315 − 0.546i)37-s − 1.25i·41-s − 1.14·43-s + (−0.290 − 0.167i)47-s + (−1.23 + 0.710i)53-s − 0.599·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.807 + 0.590i$
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: \((2,\ 1764,\ (\ :1),\ -0.807 + 0.590i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3243205986\)
\(L(\frac12)\) \(\approx\) \(0.3243205986\)
\(L(2)\) 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 \( 1 + (-3.08 - 1.77i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.03 - 4.63i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 4.24T + 169T^{2} \)
17 \( 1 + (-2.11 + 1.22i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (10.3 - 17.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (0.682 + 0.394i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 7.69iT - 841T^{2} \)
31 \( 1 + (11.7 + 20.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.6 + 20.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 51.5iT - 1.68e3T^{2} \)
43 \( 1 + 49.3T + 1.84e3T^{2} \)
47 \( 1 + (13.6 + 7.88i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (65.1 - 37.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-13.6 + 7.88i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-4.01 + 6.94i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-43.6 - 75.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 40.0iT - 5.04e3T^{2} \)
73 \( 1 + (17.2 + 29.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (4.34 - 7.52i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 115. iT - 6.88e3T^{2} \)
89 \( 1 + (81.6 + 47.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 154.T + 9.40e3T^{2} \)
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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

−8.809688946359754768465573138618, −7.935816513999903564231888498370, −7.27656396149500615403437300863, −6.29078584700937632443896116158, −5.65655750727733504082625400347, −4.72262918760174831533670069598, −3.71689715481776167850344268952, −2.55102194421911945454253207204, −1.80274517299070947311931120747, −0.079165499004023745110933366916, 1.36059618670705872234193765402, 2.48435231530541300103968199261, 3.42151078991878426411096930851, 4.74444390356538990695705231308, 5.26102376287809852269206794973, 6.21580454332221423700589592629, 6.98086951173731841000764235804, 7.991606710986368060713963422915, 8.611203004267529632498791210139, 9.505831219398485347197334199163

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