Properties

Label 2-42e2-7.5-c2-0-9
Degree $2$
Conductor $1764$
Sign $0.982 - 0.188i$
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  + (−4.65 − 2.68i)5-s + (−4.29 − 7.44i)11-s + 21.0i·13-s + (−4.75 + 2.74i)17-s + (−6.27 − 3.62i)19-s + (14.0 − 24.2i)23-s + (1.97 + 3.41i)25-s − 40.3·29-s + (35.0 − 20.2i)31-s + (−33.3 + 57.7i)37-s + 33.6i·41-s + 0.932·43-s + (74.1 + 42.8i)47-s + (−22.2 − 38.6i)53-s + 46.2i·55-s + ⋯
L(s)  = 1  + (−0.931 − 0.537i)5-s + (−0.390 − 0.676i)11-s + 1.61i·13-s + (−0.279 + 0.161i)17-s + (−0.330 − 0.190i)19-s + (0.609 − 1.05i)23-s + (0.0788 + 0.136i)25-s − 1.39·29-s + (1.13 − 0.653i)31-s + (−0.900 + 1.55i)37-s + 0.820i·41-s + 0.0216·43-s + (1.57 + 0.911i)47-s + (−0.420 − 0.728i)53-s + 0.840i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.982 - 0.188i$
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: \((2,\ 1764,\ (\ :1),\ 0.982 - 0.188i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.209497455\)
\(L(\frac12)\) \(\approx\) \(1.209497455\)
\(L(2)\) 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 \( 1 + (4.65 + 2.68i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (4.29 + 7.44i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.0iT - 169T^{2} \)
17 \( 1 + (4.75 - 2.74i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (6.27 + 3.62i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-14.0 + 24.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 40.3T + 841T^{2} \)
31 \( 1 + (-35.0 + 20.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (33.3 - 57.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 33.6iT - 1.68e3T^{2} \)
43 \( 1 - 0.932T + 1.84e3T^{2} \)
47 \( 1 + (-74.1 - 42.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (22.2 + 38.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (55.1 - 31.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.7 - 16.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-23.8 - 41.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 14.9T + 5.04e3T^{2} \)
73 \( 1 + (-121. + 70.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-61.1 + 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 33.1iT - 6.88e3T^{2} \)
89 \( 1 + (31.2 + 18.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 16.2iT - 9.40e3T^{2} \)
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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

−8.913544807474663846573284548270, −8.460984311121314039471732081301, −7.65694863828606713658207872640, −6.74828788954118132960310889136, −6.02951537274194139985354812965, −4.71938133617049102242942029233, −4.34233872011327979173389498630, −3.27261482367572828328189716885, −2.06237681351871475703748067688, −0.67410453183815055571797847099, 0.51422667320134399295908825352, 2.14181469817116752872774305573, 3.25313650428209334624663794733, 3.88229588657198573118537506093, 5.08783738089846064800456225961, 5.72726226351639799696903854374, 7.01243422562012853333479454742, 7.47710648406667694890222152216, 8.111295371278554455645163674914, 9.059755390142747291107066954669

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