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Properties

Label 2-1620-45.38-c1-0-18
Degree $2$
Conductor $1620$
Sign $0.732 + 0.680i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.91 + 1.15i)5-s + (−0.814 − 3.03i)7-s + (4.91 + 2.83i)11-s + (0.448 − 1.67i)13-s + (−5.67 − 5.67i)17-s − 5.89i·19-s + (1.74 + 0.467i)23-s + (2.32 + 4.42i)25-s + (2.83 − 4.91i)29-s + (2.22 + 3.85i)31-s + (1.95 − 6.75i)35-s + (−3.67 + 3.67i)37-s + (7.12 − 4.11i)41-s + (−7.44 + 1.99i)43-s + (−2.51 + 1.44i)49-s + ⋯
L(s)  = 1  + (0.855 + 0.517i)5-s + (−0.307 − 1.14i)7-s + (1.48 + 0.856i)11-s + (0.124 − 0.464i)13-s + (−1.37 − 1.37i)17-s − 1.35i·19-s + (0.363 + 0.0974i)23-s + (0.464 + 0.885i)25-s + (0.527 − 0.913i)29-s + (0.399 + 0.692i)31-s + (0.331 − 1.14i)35-s + (−0.604 + 0.604i)37-s + (1.11 − 0.642i)41-s + (−1.13 + 0.304i)43-s + (−0.358 + 0.207i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.732 + 0.680i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.732 + 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.005128869\)
\(L(\frac12)\) \(\approx\) \(2.005128869\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.91 - 1.15i)T \)
good7 \( 1 + (0.814 + 3.03i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-4.91 - 2.83i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.448 + 1.67i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.67 + 5.67i)T + 17iT^{2} \)
19 \( 1 + 5.89iT - 19T^{2} \)
23 \( 1 + (-1.74 - 0.467i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.83 + 4.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.22 - 3.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.67 - 3.67i)T - 37iT^{2} \)
41 \( 1 + (-7.12 + 4.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.44 - 1.99i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.27 + 1.27i)T - 53iT^{2} \)
59 \( 1 + (-1.27 - 2.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.39 + 5.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.81 + 0.485i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.23iT - 71T^{2} \)
73 \( 1 + (3.77 + 3.77i)T + 73iT^{2} \)
79 \( 1 + (-13.2 - 7.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.54 - 9.50i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.55T + 89T^{2} \)
97 \( 1 + (2.64 + 9.86i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.442059536423654834882896571768, −8.751183188849719313878423110398, −7.34718982275297058383566288482, −6.78622275083410116143822757237, −6.47228751021697278638593551762, −5.01592769034848633996820211882, −4.34911313219323084519630863351, −3.22229534000171449075294863685, −2.19628296028529705161086035252, −0.842720202609618851195842107020, 1.38301669494387671652638846484, 2.24656736428105870983265359516, 3.56998078627187590815741661508, 4.46305460816715146466541598719, 5.69162176914225085329072840158, 6.15574015731786033067577329634, 6.72073181631214931175269448244, 8.329758515077678769475945526856, 8.800087879892808240355256415257, 9.229390867431695309672512393658

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