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Properties

Label 2-1620-60.47-c0-0-3
Degree $2$
Conductor $1620$
Sign $-0.685 + 0.727i$
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 + (−0.258 − 0.965i)5-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.500i)10-s + (1.36 − 1.36i)13-s − 1.00·16-s + (−1.22 + 1.22i)17-s + (−0.965 + 0.258i)20-s + (−0.866 + 0.499i)25-s − 1.93i·26-s + 0.517·29-s + (−0.707 + 0.707i)32-s + 1.73i·34-s + (−0.366 − 0.366i)37-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.258 − 0.965i)5-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.500i)10-s + (1.36 − 1.36i)13-s − 1.00·16-s + (−1.22 + 1.22i)17-s + (−0.965 + 0.258i)20-s + (−0.866 + 0.499i)25-s − 1.93i·26-s + 0.517·29-s + (−0.707 + 0.707i)32-s + 1.73i·34-s + (−0.366 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.685 + 0.727i$
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: \((2,\ 1620,\ (\ :0),\ -0.685 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.466516571\)
\(L(\frac12)\) \(\approx\) \(1.466516571\)
\(L(1)\) 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 \( 1 + (0.258 + 0.965i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 0.517T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
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 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.93T + T^{2} \)
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

−9.219236774377465034741024223827, −8.657763602076204048071120822635, −7.916908006841044297481760186517, −6.55631340333004375664980792855, −5.81783058493684125184039301571, −5.09581389565937010349993109616, −4.08679409073861427608742218626, −3.51859965101872483672066882489, −2.13995327480068837656044663596, −0.956148891287532981728575777598, 2.19390629362508981274093439500, 3.23260049113057863805176944408, 4.08560928084775992775442789835, 4.85116583072543224897640440412, 6.07877906700969303633415754468, 6.70613549832146086268397994465, 7.10951242587744901535961933587, 8.223429885019670373969688603657, 8.848385817632453182780527671625, 9.747996591352797232682278875238

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