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Properties

Label 2-1620-45.32-c1-0-10
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: $\chi_{1620} (377, \cdot )$
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

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

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