Properties

Label 2-42e2-7.3-c2-0-3
Degree $2$
Conductor $1764$
Sign $-0.580 - 0.814i$
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  + (4.91 − 2.83i)5-s + (−7.94 + 13.7i)11-s − 20.1i·13-s + (0.823 + 0.475i)17-s + (−27.5 + 15.9i)19-s + (13 + 22.5i)23-s + (3.60 − 6.23i)25-s − 27.7·29-s + (−13.1 − 7.57i)31-s + (16 + 27.7i)37-s + 17.3i·41-s − 59.2·43-s + (−66.1 + 38.1i)47-s + (12.8 − 22.3i)53-s + 90.2i·55-s + ⋯
L(s)  = 1  + (0.982 − 0.567i)5-s + (−0.722 + 1.25i)11-s − 1.55i·13-s + (0.0484 + 0.0279i)17-s + (−1.45 + 0.838i)19-s + (0.565 + 0.978i)23-s + (0.144 − 0.249i)25-s − 0.958·29-s + (−0.423 − 0.244i)31-s + (0.432 + 0.748i)37-s + 0.422i·41-s − 1.37·43-s + (−1.40 + 0.812i)47-s + (0.243 − 0.421i)53-s + 1.64i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.580 - 0.814i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8357363738\)
\(L(\frac12)\) \(\approx\) \(0.8357363738\)
\(L(2)\) 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 \( 1 + (-4.91 + 2.83i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.94 - 13.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 20.1iT - 169T^{2} \)
17 \( 1 + (-0.823 - 0.475i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (27.5 - 15.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13 - 22.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 27.7T + 841T^{2} \)
31 \( 1 + (13.1 + 7.57i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-16 - 27.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 17.3iT - 1.68e3T^{2} \)
43 \( 1 + 59.2T + 1.84e3T^{2} \)
47 \( 1 + (66.1 - 38.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-12.8 + 22.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-59.2 - 34.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-46.9 + 27.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (43.6 - 75.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.4T + 5.04e3T^{2} \)
73 \( 1 + (-60.9 - 35.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (20.1 + 34.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 71.5iT - 6.88e3T^{2} \)
89 \( 1 + (66.9 - 38.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 128. iT - 9.40e3T^{2} \)
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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

−9.673610699828253289983264393486, −8.521054795595783690262471689110, −7.915814707408871952144781626338, −7.05486181165001048653453339413, −5.96183897223752489717683843462, −5.38100253574697472617338035795, −4.66302027174591465749856452071, −3.42774675229964699049481313543, −2.26659924088150940239175126674, −1.43281640718091463804636531235, 0.19721053128674291811314945057, 1.90119443438419738761162314692, 2.57962012607466835542330736112, 3.71981932731047642782672522503, 4.79898758667429711355187445539, 5.70241464457058223504633160280, 6.54194435217873537202607022051, 6.92252347110782626203427819654, 8.248542584954082176956312687313, 8.860594984619361917854777209202

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