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Properties

Label 2-1920-20.7-c1-0-46
Degree $2$
Conductor $1920$
Sign $-0.821 + 0.570i$
Analytic cond. $15.3312$
Root an. cond. $3.91551$
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  + (−0.707 − 0.707i)3-s + (−0.118 − 2.23i)5-s + (2.68 − 2.68i)7-s + 1.00i·9-s − 5.80i·11-s + (2.15 − 2.15i)13-s + (−1.49 + 1.66i)15-s + (−3.88 − 3.88i)17-s + 4.64·19-s − 3.80·21-s + (2.62 + 2.62i)23-s + (−4.97 + 0.530i)25-s + (0.707 − 0.707i)27-s − 5.95i·29-s + 6.89i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.0531 − 0.998i)5-s + (1.01 − 1.01i)7-s + 0.333i·9-s − 1.74i·11-s + (0.598 − 0.598i)13-s + (−0.385 + 0.429i)15-s + (−0.941 − 0.941i)17-s + 1.06·19-s − 0.830·21-s + (0.546 + 0.546i)23-s + (−0.994 + 0.106i)25-s + (0.136 − 0.136i)27-s − 1.10i·29-s + 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.821 + 0.570i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.684414610\)
\(L(\frac12)\) \(\approx\) \(1.684414610\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.118 + 2.23i)T \)
good7 \( 1 + (-2.68 + 2.68i)T - 7iT^{2} \)
11 \( 1 + 5.80iT - 11T^{2} \)
13 \( 1 + (-2.15 + 2.15i)T - 13iT^{2} \)
17 \( 1 + (3.88 + 3.88i)T + 17iT^{2} \)
19 \( 1 - 4.64T + 19T^{2} \)
23 \( 1 + (-2.62 - 2.62i)T + 23iT^{2} \)
29 \( 1 + 5.95iT - 29T^{2} \)
31 \( 1 - 6.89iT - 31T^{2} \)
37 \( 1 + (-1.49 - 1.49i)T + 37iT^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + (-6.04 - 6.04i)T + 43iT^{2} \)
47 \( 1 + (2.98 - 2.98i)T - 47iT^{2} \)
53 \( 1 + (6.19 - 6.19i)T - 53iT^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (-4.10 + 4.10i)T - 67iT^{2} \)
71 \( 1 - 2.74iT - 71T^{2} \)
73 \( 1 + (0.673 - 0.673i)T - 73iT^{2} \)
79 \( 1 - 9.89T + 79T^{2} \)
83 \( 1 + (2.42 + 2.42i)T + 83iT^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + (6.48 + 6.48i)T + 97iT^{2} \)
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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

−8.811721800126228752613954643645, −7.895994687661105234561733436106, −7.65534312530933447794370860623, −6.40996931164582218593053171037, −5.58791060950237259035829214871, −4.91146098213925153701035035262, −4.06201599691719352930018209264, −2.91623432588624992215378099985, −1.20151694911084539878388180446, −0.76614834787987349693934757280, 1.77506777997159829716945502591, 2.50238346324690198487879346679, 3.89981429401851571866106264135, 4.61616617829629060929018591117, 5.49134869508204195726202833026, 6.34842740012063055306146141123, 7.11356276347469876476181861151, 7.86944973549848869463193022347, 8.903133041152047893381606097615, 9.480019021241033486993805476619

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