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Properties

Label 2-1620-180.139-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.642 - 0.766i$
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  + i·2-s − 4-s + (−0.866 − 0.5i)5-s − i·8-s + (0.5 − 0.866i)10-s + 16-s + i·17-s + 1.73i·19-s + (0.866 + 0.5i)20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (1.5 + 0.866i)31-s + i·32-s − 34-s − 1.73·38-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−0.866 − 0.5i)5-s − i·8-s + (0.5 − 0.866i)10-s + 16-s + i·17-s + 1.73i·19-s + (0.866 + 0.5i)20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (1.5 + 0.866i)31-s + i·32-s − 34-s − 1.73·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.642 - 0.766i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7062277475\)
\(L(\frac12)\) \(\approx\) \(0.7062277475\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.787245989920310925715837437976, −8.740672841455480646307892123629, −8.119570550924413015781471138734, −7.71023553136310833600801246462, −6.67641740181480838196053470600, −5.83793345669545772773312487639, −5.07525055456791805950029516436, −4.01684760411957592133259578399, −3.53447349436593012891692653401, −1.43622322828939453511757796115, 0.60702152034050772066015842305, 2.49798537604378542099073949450, 2.99412046526529264398666233906, 4.35306327982578892167889707513, 4.62437775692355609034279256720, 6.01801089571756233961904196346, 7.02336254890689043658756510105, 7.85384873513351547399084347218, 8.655023367566095781886980157318, 9.357788791198430945951492258132

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