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Properties

Label 2-1620-45.29-c2-0-46
Degree $2$
Conductor $1620$
Sign $-0.993 - 0.114i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (4.99 − 0.299i)5-s + (−6.92 − 4i)7-s + (−7.74 − 4.47i)11-s + (−10.3 + 6i)13-s + 31.3·17-s − 6·19-s + (2.23 + 3.87i)23-s + (24.8 − 2.99i)25-s + (−23.2 − 13.4i)29-s + (−17 − 29.4i)31-s + (−35.7 − 17.8i)35-s + 44i·37-s + (−15.4 + 8.94i)41-s + (−24.2 − 14i)43-s + (2.23 − 3.87i)47-s + ⋯
L(s)  = 1  + (0.998 − 0.0599i)5-s + (−0.989 − 0.571i)7-s + (−0.704 − 0.406i)11-s + (−0.799 + 0.461i)13-s + 1.84·17-s − 0.315·19-s + (0.0972 + 0.168i)23-s + (0.992 − 0.119i)25-s + (−0.801 − 0.462i)29-s + (−0.548 − 0.949i)31-s + (−1.02 − 0.511i)35-s + 1.18i·37-s + (−0.377 + 0.218i)41-s + (−0.563 − 0.325i)43-s + (0.0475 − 0.0824i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.993 - 0.114i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.993 - 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1767392370\)
\(L(\frac12)\) \(\approx\) \(0.1767392370\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-4.99 + 0.299i)T \)
good7 \( 1 + (6.92 + 4i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (7.74 + 4.47i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (10.3 - 6i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 31.3T + 289T^{2} \)
19 \( 1 + 6T + 361T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (23.2 + 13.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (17 + 29.4i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 44iT - 1.36e3T^{2} \)
41 \( 1 + (15.4 - 8.94i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (24.2 + 14i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-2.23 + 3.87i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 40.2T + 2.80e3T^{2} \)
59 \( 1 + (85.2 - 49.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37 - 64.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (79.6 - 46i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 53.6iT - 5.04e3T^{2} \)
73 \( 1 + 56iT - 5.32e3T^{2} \)
79 \( 1 + (-39 + 67.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (51.4 - 89.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 + (-27.7 - 16i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.007522618459062349609877151934, −7.86542151827798438589074356778, −7.22410704384483297092551140435, −6.23235522340959448165608937290, −5.65745491146175735402510735992, −4.73325785795928021054293431154, −3.48309795429317477501662916643, −2.71331063725859911847606084985, −1.45491846827417809694650204337, −0.04421477625999639498248510872, 1.63918984811085404407385668628, 2.74083283639085502242290794595, 3.37962813660079851686207312879, 5.00279041761613456916403475094, 5.51568360148776502004034489635, 6.27964509234793826456495959611, 7.21583133347051003892759301859, 7.956271069610686030469306428813, 9.116275197304157166450488097872, 9.606480775780881686971000627522

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