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Properties

Label 2-1920-80.69-c1-0-37
Degree $2$
Conductor $1920$
Sign $0.606 + 0.794i$
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 + (2.09 − 0.770i)5-s + 3.05·7-s + 1.00i·9-s + (1.80 + 1.80i)11-s + (−2.47 − 2.47i)13-s + (−2.02 − 0.939i)15-s − 3.66i·17-s + (2.31 − 2.31i)19-s + (−2.15 − 2.15i)21-s + 4.86·23-s + (3.81 − 3.23i)25-s + (0.707 − 0.707i)27-s + (−4.74 + 4.74i)29-s + 1.86·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.938 − 0.344i)5-s + 1.15·7-s + 0.333i·9-s + (0.545 + 0.545i)11-s + (−0.685 − 0.685i)13-s + (−0.523 − 0.242i)15-s − 0.889i·17-s + (0.531 − 0.531i)19-s + (−0.470 − 0.470i)21-s + 1.01·23-s + (0.762 − 0.646i)25-s + (0.136 − 0.136i)27-s + (−0.881 + 0.881i)29-s + 0.334·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $0.606 + 0.794i$
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: \((2,\ 1920,\ (\ :1/2),\ 0.606 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.145964324\)
\(L(\frac12)\) \(\approx\) \(2.145964324\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.09 + 0.770i)T \)
good7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 + (-1.80 - 1.80i)T + 11iT^{2} \)
13 \( 1 + (2.47 + 2.47i)T + 13iT^{2} \)
17 \( 1 + 3.66iT - 17T^{2} \)
19 \( 1 + (-2.31 + 2.31i)T - 19iT^{2} \)
23 \( 1 - 4.86T + 23T^{2} \)
29 \( 1 + (4.74 - 4.74i)T - 29iT^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 + (5.40 - 5.40i)T - 37iT^{2} \)
41 \( 1 + 6.47iT - 41T^{2} \)
43 \( 1 + (-4.19 + 4.19i)T - 43iT^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 + (-9.99 + 9.99i)T - 53iT^{2} \)
59 \( 1 + (-2.47 - 2.47i)T + 59iT^{2} \)
61 \( 1 + (8.01 - 8.01i)T - 61iT^{2} \)
67 \( 1 + (-8.60 - 8.60i)T + 67iT^{2} \)
71 \( 1 + 6.63iT - 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (3.65 + 3.65i)T + 83iT^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−9.086702719853888794928536203233, −8.374750445881069367968028080126, −7.20512995532573208613892258875, −7.01892009709952946545298108358, −5.60678883873619469160552401706, −5.19315123203809178331565150245, −4.50763082545698926567328836813, −2.89633121964366713994402949425, −1.88678889961891569721919777332, −0.952900691612522634343624137130, 1.30053442403316641829734612360, 2.24121345974579655915954528055, 3.55749204650739453408930536076, 4.52724867886022201092690834351, 5.35348769052648474403165140120, 5.99897178963571730372556546954, 6.85321699337094037912246230412, 7.72734614917491580072915298039, 8.704583792714522275101320579012, 9.367546119871786683699002406310

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