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Properties

Label 2-42e2-7.5-c2-0-0
Degree $2$
Conductor $1764$
Sign $-0.937 - 0.347i$
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  + (5.04 + 2.91i)5-s + (−3.43 − 5.94i)11-s − 3.62i·13-s + (−8.41 + 4.85i)17-s + (−26.2 − 15.1i)19-s + (−9.07 + 15.7i)23-s + (4.48 + 7.76i)25-s + 40.4·29-s + (−47.8 + 27.6i)31-s + (−27.4 + 47.6i)37-s + 56.3i·41-s − 66.0·43-s + (42.8 + 24.7i)47-s + (−40.5 − 70.1i)53-s − 39.9i·55-s + ⋯
L(s)  = 1  + (1.00 + 0.582i)5-s + (−0.311 − 0.540i)11-s − 0.278i·13-s + (−0.494 + 0.285i)17-s + (−1.38 − 0.797i)19-s + (−0.394 + 0.683i)23-s + (0.179 + 0.310i)25-s + 1.39·29-s + (−1.54 + 0.890i)31-s + (−0.742 + 1.28i)37-s + 1.37i·41-s − 1.53·43-s + (0.910 + 0.525i)47-s + (−0.764 − 1.32i)53-s − 0.727i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6309579098\)
\(L(\frac12)\) \(\approx\) \(0.6309579098\)
\(L(2)\) 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 \( 1 + (-5.04 - 2.91i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.43 + 5.94i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 3.62iT - 169T^{2} \)
17 \( 1 + (8.41 - 4.85i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (26.2 + 15.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (9.07 - 15.7i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 40.4T + 841T^{2} \)
31 \( 1 + (47.8 - 27.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (27.4 - 47.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 56.3iT - 1.68e3T^{2} \)
43 \( 1 + 66.0T + 1.84e3T^{2} \)
47 \( 1 + (-42.8 - 24.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (40.5 + 70.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (30.1 - 17.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-0.0331 - 0.0191i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-32.0 - 55.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 50.2T + 5.04e3T^{2} \)
73 \( 1 + (-18.4 + 10.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-23.7 + 41.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 33.6iT - 6.88e3T^{2} \)
89 \( 1 + (-135. - 78.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 43.7iT - 9.40e3T^{2} \)
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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

−9.514849691316652848974006069918, −8.653881409677197984496293680926, −8.036861378565084268134385888329, −6.72902006266275814962223990911, −6.45845480144055358288057835514, −5.47772194319735899813746835812, −4.65300188809522659265249453541, −3.38792774218030366088526732419, −2.53838851101818178671711980980, −1.55359913307349186576507165125, 0.14569096854612154654395077733, 1.77606015377244624829886917859, 2.30887256003744516884899628342, 3.84368617998363385705916741684, 4.69061976142935953307069792072, 5.53934212549803689944279966994, 6.27373381409881135311111015107, 7.09527987716219174801790852732, 8.097546333106227170454214655836, 8.920182543307710199732501930577

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