Properties

Label 2-1620-180.139-c0-0-7
Degree $2$
Conductor $1620$
Sign $0.342 - 0.939i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s i·17-s − 1.73i·19-s + 0.999i·20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s + (0.866 − 1.49i)38-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.866 + 0.5i)5-s + 0.999i·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s i·17-s − 1.73i·19-s + 0.999i·20-s + (−0.866 + 1.5i)23-s + (0.499 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s + (0.5 − 0.866i)34-s + (0.866 − 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.342 - 0.939i$
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: \((2,\ 1620,\ (\ :0),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.059693657\)
\(L(\frac12)\) \(\approx\) \(2.059693657\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
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

−9.488478737050878408180406797730, −9.123824670669407428140685423797, −7.80459747186206550250283137070, −7.17984087081652675624715232000, −6.49160182955838938500626798135, −5.57197238257233679096340628042, −5.04470615060909331958343355146, −3.86289823568665006911219669556, −2.88976221028055071545930440726, −2.02431055089821508885389727722, 1.48230050296191406741310029786, 2.26242527132989234558910948549, 3.57255283615605177361288474575, 4.35764931623525485525467436163, 5.36737219805671584897445819349, 5.99184138148652764590891469838, 6.62642737977617720361907892094, 7.88588828543812388200220977461, 8.766064760443898842999176615245, 9.641281582971000526324660776285

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