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Properties

Label 2-1620-1620.1159-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.952 + 0.305i$
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  + (0.893 − 0.448i)2-s + (0.973 − 0.230i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)5-s + (0.766 − 0.642i)6-s + (−0.543 + 1.26i)7-s + (0.173 − 0.984i)8-s + (0.893 − 0.448i)9-s + (0.173 + 0.984i)10-s + (0.396 − 0.918i)12-s + (0.0798 + 1.37i)14-s + (−0.0581 + 0.998i)15-s + (−0.286 − 0.957i)16-s + (0.597 − 0.802i)18-s + (0.597 + 0.802i)20-s + (−0.238 + 1.35i)21-s + ⋯
L(s)  = 1  + (0.893 − 0.448i)2-s + (0.973 − 0.230i)3-s + (0.597 − 0.802i)4-s + (−0.286 + 0.957i)5-s + (0.766 − 0.642i)6-s + (−0.543 + 1.26i)7-s + (0.173 − 0.984i)8-s + (0.893 − 0.448i)9-s + (0.173 + 0.984i)10-s + (0.396 − 0.918i)12-s + (0.0798 + 1.37i)14-s + (−0.0581 + 0.998i)15-s + (−0.286 − 0.957i)16-s + (0.597 − 0.802i)18-s + (0.597 + 0.802i)20-s + (−0.238 + 1.35i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.952 + 0.305i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.952 + 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.352636747\)
\(L(\frac12)\) \(\approx\) \(2.352636747\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.893 + 0.448i)T \)
3 \( 1 + (-0.973 + 0.230i)T \)
5 \( 1 + (0.286 - 0.957i)T \)
good7 \( 1 + (0.543 - 1.26i)T + (-0.686 - 0.727i)T^{2} \)
11 \( 1 + (-0.893 - 0.448i)T^{2} \)
13 \( 1 + (-0.396 + 0.918i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
19 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.313 - 0.727i)T + (-0.686 + 0.727i)T^{2} \)
29 \( 1 + (-0.115 + 1.98i)T + (-0.993 - 0.116i)T^{2} \)
31 \( 1 + (-0.973 - 0.230i)T^{2} \)
37 \( 1 + (-0.173 + 0.984i)T^{2} \)
41 \( 1 + (1.49 + 0.749i)T + (0.597 + 0.802i)T^{2} \)
43 \( 1 + (0.238 - 0.252i)T + (-0.0581 - 0.998i)T^{2} \)
47 \( 1 + (1.77 - 0.207i)T + (0.973 - 0.230i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.893 + 0.448i)T^{2} \)
61 \( 1 + (-1.16 - 1.56i)T + (-0.286 + 0.957i)T^{2} \)
67 \( 1 + (-0.00676 - 0.116i)T + (-0.993 + 0.116i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (-0.597 + 0.802i)T^{2} \)
83 \( 1 + (1.49 - 0.749i)T + (0.597 - 0.802i)T^{2} \)
89 \( 1 + (-0.310 + 1.76i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (0.835 - 0.549i)T^{2} \)
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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

−9.809664170966041088139258258624, −8.815362451431986169686464290726, −7.893378759067471220610421418840, −6.97121637285340915627373714549, −6.33075977232696440030657207691, −5.49939963974668420767845406565, −4.23033922900061203268762042398, −3.33678312628836428464124156249, −2.72945393974726244616442938738, −1.92931860385832624042396779901, 1.60676390013566696565705723484, 3.15538191755864473084106396586, 3.71504493613594269064667621308, 4.60819037694498987685929021185, 5.16406667306569522853336078514, 6.67795852867886311923542724154, 7.09713169021175247301123500800, 8.107142044232654436012239708052, 8.525983901465902329381562906417, 9.515455603898998516168373730782

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