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Properties

Label 2-1920-80.69-c1-0-15
Degree $2$
Conductor $1920$
Sign $-0.0794 - 0.996i$
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 + (2.23 − 0.162i)5-s − 2.93·7-s + 1.00i·9-s + (0.663 + 0.663i)11-s + (1.12 + 1.12i)13-s + (1.69 + 1.46i)15-s + 7.47i·17-s + (−0.423 + 0.423i)19-s + (−2.07 − 2.07i)21-s − 6.17·23-s + (4.94 − 0.722i)25-s + (−0.707 + 0.707i)27-s + (2.95 − 2.95i)29-s + 1.82·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.997 − 0.0724i)5-s − 1.10·7-s + 0.333i·9-s + (0.200 + 0.200i)11-s + (0.312 + 0.312i)13-s + (0.436 + 0.377i)15-s + 1.81i·17-s + (−0.0971 + 0.0971i)19-s + (−0.453 − 0.453i)21-s − 1.28·23-s + (0.989 − 0.144i)25-s + (−0.136 + 0.136i)27-s + (0.548 − 0.548i)29-s + 0.327·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.0794 - 0.996i$
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: \((2,\ 1920,\ (\ :1/2),\ -0.0794 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862085175\)
\(L(\frac12)\) \(\approx\) \(1.862085175\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.162i)T \)
good7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + (-0.663 - 0.663i)T + 11iT^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 - 7.47iT - 17T^{2} \)
19 \( 1 + (0.423 - 0.423i)T - 19iT^{2} \)
23 \( 1 + 6.17T + 23T^{2} \)
29 \( 1 + (-2.95 + 2.95i)T - 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (5.53 - 5.53i)T - 37iT^{2} \)
41 \( 1 - 12.3iT - 41T^{2} \)
43 \( 1 + (-0.897 + 0.897i)T - 43iT^{2} \)
47 \( 1 + 4.12iT - 47T^{2} \)
53 \( 1 + (-0.146 + 0.146i)T - 53iT^{2} \)
59 \( 1 + (-7.72 - 7.72i)T + 59iT^{2} \)
61 \( 1 + (-7.37 + 7.37i)T - 61iT^{2} \)
67 \( 1 + (-8.68 - 8.68i)T + 67iT^{2} \)
71 \( 1 - 8.95iT - 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (9.18 + 9.18i)T + 83iT^{2} \)
89 \( 1 + 8.71iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−9.641489217632353303975644763484, −8.599962821555353745548314109027, −8.182688090751358619453595950949, −6.73048267469512933247262009121, −6.30715131781762041620120884486, −5.54345372995107782961167747675, −4.33066141215574564271021359059, −3.59851046489455144359081921851, −2.54361782323055193962206568451, −1.53616711662553245925283723661, 0.62410249217682780935471586169, 2.10068370212679522111568815010, 2.90915041299655580709371642022, 3.76902572703992516238535172036, 5.14708489770375573552789075005, 5.89863550463911769809700062266, 6.70758357857257898146795762761, 7.20109580494940996718075333043, 8.348701753061105308289198081218, 9.125622819804190571672677869594

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