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Properties

Label 2-1620-45.23-c1-0-18
Degree $2$
Conductor $1620$
Sign $0.619 + 0.784i$
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  + (1.11 − 1.93i)5-s + (1.36 + 0.366i)7-s + (3.87 − 2.23i)11-s + (4.09 − 1.09i)13-s + (−2.23 − 2.23i)17-s + 2i·19-s + (0.818 + 3.05i)23-s + (−2.5 − 4.33i)25-s + (−2.23 − 3.87i)29-s + (−2 + 3.46i)31-s + (2.23 − 2.23i)35-s + (−3 + 3i)37-s + (7.74 + 4.47i)41-s + (−1.09 + 4.09i)43-s + (2.45 − 9.16i)47-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)5-s + (0.516 + 0.138i)7-s + (1.16 − 0.674i)11-s + (1.13 − 0.304i)13-s + (−0.542 − 0.542i)17-s + 0.458i·19-s + (0.170 + 0.636i)23-s + (−0.5 − 0.866i)25-s + (−0.415 − 0.719i)29-s + (−0.359 + 0.622i)31-s + (0.377 − 0.377i)35-s + (−0.493 + 0.493i)37-s + (1.20 + 0.698i)41-s + (−0.167 + 0.624i)43-s + (0.358 − 1.33i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.619 + 0.784i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.619 + 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.194326271\)
\(L(\frac12)\) \(\approx\) \(2.194326271\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 1.93i)T \)
good7 \( 1 + (-1.36 - 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.87 + 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.09 + 1.09i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.23 + 2.23i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-0.818 - 3.05i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.23 + 3.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (-7.74 - 4.47i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.09 - 4.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.45 + 9.16i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.23 - 2.23i)T - 53iT^{2} \)
59 \( 1 + (-4.47 + 7.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + (-5.19 + 3i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.16 - 2.45i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + (12.2 + 3.29i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.164960803391611544349103936345, −8.578401422843794311268921799162, −7.943393485501290368325421317343, −6.70205836803175755402379496388, −5.96562075081514679034929072956, −5.24742461797018143397772414459, −4.25729639419680885511472570736, −3.38271765178382891903760379679, −1.87003445935984427014040164447, −0.976486658054762368523879259690, 1.41878981870276321599380971754, 2.33298569583198620094557995712, 3.67267182813939585496174811113, 4.32580503661713705091055593264, 5.55785502739235535804502491653, 6.44546313509267821857791311904, 6.91934497188097447967240951761, 7.84541116236569198392343516934, 8.967759013186351450833448593481, 9.306489028361316967659268972967

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