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Properties

Label 2-1920-20.3-c1-0-0
Degree $2$
Conductor $1920$
Sign $-0.989 - 0.141i$
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 + (0.893 + 2.04i)5-s + (−0.804 − 0.804i)7-s − 1.00i·9-s + 3.13i·11-s + (−3.89 − 3.89i)13-s + (2.08 + 0.817i)15-s + (−5.24 + 5.24i)17-s − 7.36·19-s − 1.13·21-s + (2.24 − 2.24i)23-s + (−3.40 + 3.66i)25-s + (−0.707 − 0.707i)27-s − 0.869i·29-s − 10.8i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.399 + 0.916i)5-s + (−0.304 − 0.304i)7-s − 0.333i·9-s + 0.946i·11-s + (−1.08 − 1.08i)13-s + (0.537 + 0.211i)15-s + (−1.27 + 1.27i)17-s − 1.68·19-s − 0.248·21-s + (0.468 − 0.468i)23-s + (−0.680 + 0.732i)25-s + (−0.136 − 0.136i)27-s − 0.161i·29-s − 1.94i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.989 - 0.141i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1715585413\)
\(L(\frac12)\) \(\approx\) \(0.1715585413\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.893 - 2.04i)T \)
good7 \( 1 + (0.804 + 0.804i)T + 7iT^{2} \)
11 \( 1 - 3.13iT - 11T^{2} \)
13 \( 1 + (3.89 + 3.89i)T + 13iT^{2} \)
17 \( 1 + (5.24 - 5.24i)T - 17iT^{2} \)
19 \( 1 + 7.36T + 19T^{2} \)
23 \( 1 + (-2.24 + 2.24i)T - 23iT^{2} \)
29 \( 1 + 0.869iT - 29T^{2} \)
31 \( 1 + 10.8iT - 31T^{2} \)
37 \( 1 + (6.85 - 6.85i)T - 37iT^{2} \)
41 \( 1 + 9.93T + 41T^{2} \)
43 \( 1 + (-1.35 + 1.35i)T - 43iT^{2} \)
47 \( 1 + (-1.96 - 1.96i)T + 47iT^{2} \)
53 \( 1 + (-3.06 - 3.06i)T + 53iT^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 - 3.07T + 61T^{2} \)
67 \( 1 + (-9.02 - 9.02i)T + 67iT^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (0.566 + 0.566i)T + 73iT^{2} \)
79 \( 1 + 6.65T + 79T^{2} \)
83 \( 1 + (6.74 - 6.74i)T - 83iT^{2} \)
89 \( 1 - 1.30iT - 89T^{2} \)
97 \( 1 + (0.0310 - 0.0310i)T - 97iT^{2} \)
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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

−9.779709789441885071486205852586, −8.657120336134811945950234928971, −8.020602613155038746190735000475, −6.98033279916477832603013506335, −6.71975313261866580200277053931, −5.78136304783408976548597978926, −4.57035324616676285395231251762, −3.70026351878494695887785774662, −2.48530490305961598572425040354, −2.02921110352241140448864420462, 0.05145317471136457727837309695, 1.85702332122907871432624671719, 2.70812511265462961813796894186, 3.92996717841502622582561504969, 4.81910371049734205256319569951, 5.33995018382316521500192487257, 6.56495539721456055054567162091, 7.11667510039901111549430075014, 8.592309411135802677546907554305, 8.746572734721391999080935919948

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