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Properties

Label 2-1620-60.47-c0-0-2
Degree $2$
Conductor $1620$
Sign $0.287 + 0.957i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.965 + 0.258i)5-s + (−0.707 − 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.366 + 0.366i)13-s − 1.00·16-s + (1.22 − 1.22i)17-s + (0.258 − 0.965i)20-s + (0.866 + 0.499i)25-s + 0.517i·26-s − 1.93·29-s + (−0.707 + 0.707i)32-s − 1.73i·34-s + (1.36 + 1.36i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.965 + 0.258i)5-s + (−0.707 − 0.707i)8-s + (0.866 − 0.500i)10-s + (−0.366 + 0.366i)13-s − 1.00·16-s + (1.22 − 1.22i)17-s + (0.258 − 0.965i)20-s + (0.866 + 0.499i)25-s + 0.517i·26-s − 1.93·29-s + (−0.707 + 0.707i)32-s − 1.73i·34-s + (1.36 + 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835444635\)
\(L(\frac12)\) \(\approx\) \(1.835444635\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 1.93T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 0.517T + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.526347982368382495597621178348, −9.145681046013186525767783842267, −7.64843756854981315282623377284, −6.86207210985360897150351491035, −5.87313414305730330649837310580, −5.35752232284933732323107723600, −4.42226148618829327593109457609, −3.25154234976711252282617464379, −2.47513300610500131550409083190, −1.37177126710068462463366644803, 1.79632143022749570785382123018, 2.99128388078274471816319378838, 3.98052729257964996945086056410, 5.00169081182262485689698191626, 5.82047529384977667205625470861, 6.15985982739030356951678551597, 7.39814420714497070466585818119, 7.925945610016242672498708175793, 8.905367794598720921178174130794, 9.601928985383910957873663173044

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