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Properties

Label 2-1920-80.69-c1-0-42
Degree $2$
Conductor $1920$
Sign $-0.522 + 0.852i$
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 + (2.06 − 0.860i)5-s − 0.707·7-s + 1.00i·9-s + (1.79 + 1.79i)11-s + (−3.86 − 3.86i)13-s + (−2.06 − 0.850i)15-s + 0.244i·17-s + (1.53 − 1.53i)19-s + (0.500 + 0.500i)21-s − 6.92·23-s + (3.51 − 3.55i)25-s + (0.707 − 0.707i)27-s + (4.89 − 4.89i)29-s − 7.60·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.922 − 0.384i)5-s − 0.267·7-s + 0.333i·9-s + (0.542 + 0.542i)11-s + (−1.07 − 1.07i)13-s + (−0.533 − 0.219i)15-s + 0.0593i·17-s + (0.352 − 0.352i)19-s + (0.109 + 0.109i)21-s − 1.44·23-s + (0.703 − 0.710i)25-s + (0.136 − 0.136i)27-s + (0.909 − 0.909i)29-s − 1.36·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(15.3312\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1920} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1920,\ (\ :1/2),\ -0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243694715\)
\(L(\frac12)\) \(\approx\) \(1.243694715\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 + 0.860i)T \)
good7 \( 1 + 0.707T + 7T^{2} \)
11 \( 1 + (-1.79 - 1.79i)T + 11iT^{2} \)
13 \( 1 + (3.86 + 3.86i)T + 13iT^{2} \)
17 \( 1 - 0.244iT - 17T^{2} \)
19 \( 1 + (-1.53 + 1.53i)T - 19iT^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + (-4.89 + 4.89i)T - 29iT^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 + (-8.47 + 8.47i)T - 37iT^{2} \)
41 \( 1 + 2.12iT - 41T^{2} \)
43 \( 1 + (0.684 - 0.684i)T - 43iT^{2} \)
47 \( 1 + 4.47iT - 47T^{2} \)
53 \( 1 + (1.47 - 1.47i)T - 53iT^{2} \)
59 \( 1 + (-5.86 - 5.86i)T + 59iT^{2} \)
61 \( 1 + (0.0537 - 0.0537i)T - 61iT^{2} \)
67 \( 1 + (7.85 + 7.85i)T + 67iT^{2} \)
71 \( 1 + 2.08iT - 71T^{2} \)
73 \( 1 - 9.69T + 73T^{2} \)
79 \( 1 - 7.34T + 79T^{2} \)
83 \( 1 + (6.80 + 6.80i)T + 83iT^{2} \)
89 \( 1 - 3.07iT - 89T^{2} \)
97 \( 1 - 1.39iT - 97T^{2} \)
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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

−9.112703937804144154492483047952, −8.016194088693432396327503966057, −7.37520481683733128472786033966, −6.43769830808778635395472226316, −5.76634346413477091465830323554, −5.06748281789202593477952244256, −4.09958903424029156964667923499, −2.67857950489633688729673728402, −1.85385128580252345528997322041, −0.46486924896711426466779163892, 1.47370433810857806732037293298, 2.61629627169980696049955456185, 3.66425381361110407773114804490, 4.66372490260668132024485359269, 5.50489455964712714815280213471, 6.36826996265727079697227585285, 6.78595975067808433906400117849, 7.88831345462460829283110824857, 8.948830658163775484594802717483, 9.695859416248689775850224765099

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