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Properties

Label 2-1920-80.69-c1-0-33
Degree $2$
Conductor $1920$
Sign $-0.0269 + 0.999i$
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 + (−0.404 − 2.19i)5-s + 1.81·7-s + 1.00i·9-s + (0.331 + 0.331i)11-s + (0.0310 + 0.0310i)13-s + (−1.26 + 1.84i)15-s + 1.00i·17-s + (2.08 − 2.08i)19-s + (−1.28 − 1.28i)21-s + 6.22·23-s + (−4.67 + 1.77i)25-s + (0.707 − 0.707i)27-s + (6.28 − 6.28i)29-s + 7.11·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.180 − 0.983i)5-s + 0.686·7-s + 0.333i·9-s + (0.100 + 0.100i)11-s + (0.00860 + 0.00860i)13-s + (−0.327 + 0.475i)15-s + 0.243i·17-s + (0.477 − 0.477i)19-s + (−0.280 − 0.280i)21-s + 1.29·23-s + (−0.934 + 0.355i)25-s + (0.136 − 0.136i)27-s + (1.16 − 1.16i)29-s + 1.27·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0269 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1920\)    =    \(2^{7} \cdot 3 \cdot 5\)
Sign: $-0.0269 + 0.999i$
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: \((2,\ 1920,\ (\ :1/2),\ -0.0269 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.554501615\)
\(L(\frac12)\) \(\approx\) \(1.554501615\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.404 + 2.19i)T \)
good7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 + (-0.331 - 0.331i)T + 11iT^{2} \)
13 \( 1 + (-0.0310 - 0.0310i)T + 13iT^{2} \)
17 \( 1 - 1.00iT - 17T^{2} \)
19 \( 1 + (-2.08 + 2.08i)T - 19iT^{2} \)
23 \( 1 - 6.22T + 23T^{2} \)
29 \( 1 + (-6.28 + 6.28i)T - 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (-0.0723 + 0.0723i)T - 37iT^{2} \)
41 \( 1 + 3.06iT - 41T^{2} \)
43 \( 1 + (3.78 - 3.78i)T - 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (7.04 - 7.04i)T - 53iT^{2} \)
59 \( 1 + (-6.68 - 6.68i)T + 59iT^{2} \)
61 \( 1 + (2.89 - 2.89i)T - 61iT^{2} \)
67 \( 1 + (-0.150 - 0.150i)T + 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (5.48 + 5.48i)T + 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−8.834063719890041831822989121417, −8.240501792707697414021176289108, −7.52862329397115420605417600048, −6.63419393740324195713358363286, −5.72809724101090745838129321815, −4.84229955958846088814231181570, −4.41392315805932895541203770558, −2.96355034279924037268449686687, −1.65608746116706746419595843372, −0.70521594308257331764467069523, 1.22331046875564955058747976635, 2.73352439899285424531310654422, 3.48973426567360182158766621273, 4.62728253411016044876275695212, 5.24968513731383914122899790618, 6.36846188359699602649445968090, 6.89875238094487023339757890192, 7.85099438471833641947606389561, 8.543611898327348573414966120399, 9.583473691523868595470331868194

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