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Properties

Label 2-2028-13.10-c1-0-9
Degree $2$
Conductor $2028$
Sign $0.997 + 0.0771i$
Analytic cond. $16.1936$
Root an. cond. $4.02413$
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  + (−0.5 − 0.866i)3-s + (−1.73 − i)7-s + (−0.499 + 0.866i)9-s + (−3 + 5.19i)17-s + (1.73 + i)19-s + 1.99i·21-s + 5·25-s + 0.999·27-s + (3 + 5.19i)29-s − 2i·31-s + (1.73 − i)37-s + (10.3 − 6i)41-s + (−2 + 3.46i)43-s + (−1.50 − 2.59i)49-s + 6·51-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.654 − 0.377i)7-s + (−0.166 + 0.288i)9-s + (−0.727 + 1.26i)17-s + (0.397 + 0.229i)19-s + 0.436i·21-s + 25-s + 0.192·27-s + (0.557 + 0.964i)29-s − 0.359i·31-s + (0.284 − 0.164i)37-s + (1.62 − 0.937i)41-s + (−0.304 + 0.528i)43-s + (−0.214 − 0.371i)49-s + 0.840·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.997 + 0.0771i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ 0.997 + 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.314599057\)
\(L(\frac12)\) \(\approx\) \(1.314599057\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good5 \( 1 - 5T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 - i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.3 + 6i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (10.3 + 6i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.3 - 6i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.66 + 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.067785166228114219736567902400, −8.337464960003849985919404473404, −7.47774666679983461991595331073, −6.70041057685122078337229628777, −6.16361606966388197049730531506, −5.20448706487560183103102399192, −4.18761138087441434411594492652, −3.27035290158419196483283478192, −2.12627547699312770981485010221, −0.860022363304645391032987063026, 0.68570461491164165898663508689, 2.48792602208548771140129886102, 3.20424758004710687034445020955, 4.40571509812581224997765160617, 5.03232292171827766474903831087, 6.02438365219837960034211202011, 6.67480813812565383867120437379, 7.52935793872837974516916624649, 8.553628086613042452022898517451, 9.361404308128479314221470579890

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