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Properties

Label 2-1620-45.14-c2-0-44
Degree $2$
Conductor $1620$
Sign $-0.882 + 0.469i$
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  + (−0.261 − 4.99i)5-s + (2.97 − 1.71i)7-s + (−11.1 + 6.41i)11-s + (−1.55 − 0.896i)13-s + 30.9·17-s − 19.2·19-s + (1.19 − 2.06i)23-s + (−24.8 + 2.61i)25-s + (31.0 − 17.9i)29-s + (20.4 − 35.4i)31-s + (−9.36 − 14.4i)35-s − 53.6i·37-s + (−2.14 − 1.24i)41-s + (−47.7 + 27.5i)43-s + (−28.5 − 49.4i)47-s + ⋯
L(s)  = 1  + (−0.0523 − 0.998i)5-s + (0.425 − 0.245i)7-s + (−1.00 + 0.582i)11-s + (−0.119 − 0.0689i)13-s + 1.82·17-s − 1.01·19-s + (0.0518 − 0.0897i)23-s + (−0.994 + 0.104i)25-s + (1.07 − 0.618i)29-s + (0.659 − 1.14i)31-s + (−0.267 − 0.411i)35-s − 1.44i·37-s + (−0.0524 − 0.0302i)41-s + (−1.11 + 0.641i)43-s + (−0.606 − 1.05i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.882 + 0.469i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.056284074\)
\(L(\frac12)\) \(\approx\) \(1.056284074\)
\(L(2)\) 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 \( 1 + (0.261 + 4.99i)T \)
good7 \( 1 + (-2.97 + 1.71i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (11.1 - 6.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.55 + 0.896i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 30.9T + 289T^{2} \)
19 \( 1 + 19.2T + 361T^{2} \)
23 \( 1 + (-1.19 + 2.06i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-31.0 + 17.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-20.4 + 35.4i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 53.6iT - 1.36e3T^{2} \)
41 \( 1 + (2.14 + 1.24i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (47.7 - 27.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (28.5 + 49.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.4T + 2.80e3T^{2} \)
59 \( 1 + (59.8 + 34.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.80 + 15.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (44.3 + 25.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 53.9iT - 5.04e3T^{2} \)
73 \( 1 - 42.7iT - 5.32e3T^{2} \)
79 \( 1 + (44.4 + 76.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (14.0 + 24.3i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 68.3iT - 7.92e3T^{2} \)
97 \( 1 + (135. - 78.0i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−8.756255732732536391063249862991, −7.910357769131846367049702921720, −7.69331432427919121777845346244, −6.34496798650934729838556843122, −5.41686566268667883053301513594, −4.77546517786312993294083385453, −3.95689831162410049933526614018, −2.63386460751797437343204897403, −1.49316643011605565696126243249, −0.28480250973555967469077203342, 1.43536412636248199850699759637, 2.81846486546002388413913562097, 3.27871295342668475745017991353, 4.66802130411617877314934551966, 5.48975896833359323388697962534, 6.34494926602929512149706862978, 7.14119828480119497993025734412, 8.122810281445442088185894578921, 8.412013565259299337805142413355, 9.777997770347361766729615277985

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