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Properties

Label 2-1620-5.4-c1-0-15
Degree $2$
Conductor $1620$
Sign $0.926 + 0.375i$
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  + (2.07 + 0.839i)5-s − 4.93i·7-s + 2.41·11-s + 2.90i·13-s + 6.86i·17-s + 4.17·19-s − 3.35i·23-s + (3.58 + 3.48i)25-s + 5.19·29-s − 6.17·31-s + (4.14 − 10.2i)35-s − 7.84i·37-s + 5.87·41-s + 4.93i·43-s − 11.9i·47-s + ⋯
L(s)  = 1  + (0.926 + 0.375i)5-s − 1.86i·7-s + 0.727·11-s + 0.806i·13-s + 1.66i·17-s + 0.958·19-s − 0.700i·23-s + (0.717 + 0.696i)25-s + 0.964·29-s − 1.10·31-s + (0.700 − 1.72i)35-s − 1.28i·37-s + 0.917·41-s + 0.752i·43-s − 1.73i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.199325916\)
\(L(\frac12)\) \(\approx\) \(2.199325916\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 0.839i)T \)
good7 \( 1 + 4.93iT - 7T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 - 2.90iT - 13T^{2} \)
17 \( 1 - 6.86iT - 17T^{2} \)
19 \( 1 - 4.17T + 19T^{2} \)
23 \( 1 + 3.35iT - 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 6.17T + 31T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 - 5.87T + 41T^{2} \)
43 \( 1 - 4.93iT - 43T^{2} \)
47 \( 1 + 11.9iT - 47T^{2} \)
53 \( 1 + 8.54iT - 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 - 9.17T + 61T^{2} \)
67 \( 1 + 4.05iT - 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 2.02iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 5.18iT - 83T^{2} \)
89 \( 1 - 3.09T + 89T^{2} \)
97 \( 1 + 0.882iT - 97T^{2} \)
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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

−9.538976752253730602590701188188, −8.607183688087704216446309287861, −7.59800642870042264248951829324, −6.78801112385441747194421426251, −6.39818680282394714395843718227, −5.24418831110442969787796373307, −4.08571961752134102895708887168, −3.61056285882823967062008888626, −2.04120237246042569486673448670, −1.05421333218261711688018379341, 1.21365738161449770588939694300, 2.48124376878164491183308734543, 3.10708088473412889446418952197, 4.78344549809447027239462562673, 5.45705715578552944519835877863, 5.94035551786512338423887855949, 6.94094256795379869146277792277, 8.004467842761623388863989891439, 8.905429830293329467313341779378, 9.391194637383967587158117790735

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