Properties

Label 2-2028-13.4-c1-0-24
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 − 4i·5-s + (1.73 − i)7-s + (−0.499 − 0.866i)9-s + (−3.46 − 2i)11-s + (−3.46 − 2i)15-s + (1 + 1.73i)17-s + (−1.73 + i)19-s − 1.99i·21-s − 11·25-s − 0.999·27-s + (3 − 5.19i)29-s − 10i·31-s + (−3.46 + 1.99i)33-s + (−4 − 6.92i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 1.78i·5-s + (0.654 − 0.377i)7-s + (−0.166 − 0.288i)9-s + (−1.04 − 0.603i)11-s + (−0.894 − 0.516i)15-s + (0.242 + 0.420i)17-s + (−0.397 + 0.229i)19-s − 0.436i·21-s − 2.20·25-s − 0.192·27-s + (0.557 − 0.964i)29-s − 1.79i·31-s + (−0.603 + 0.348i)33-s + (−0.676 − 1.17i)35-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: $\chi_{2028} (1837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ -0.997 + 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.520115856\)
\(L(\frac12)\) \(\approx\) \(1.520115856\)
\(L(\frac{3}{2})\) 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 \( 1 + 4iT - 5T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.46 + 2i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
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 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (-8.66 - 5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.92 + 4i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.8 + 8i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (3.46 + 2i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.390800613113172570096847397182, −8.171763259670817354731782034486, −7.61830472450943018245564035986, −6.20094730782170497679266898390, −5.55084746919041315040853094391, −4.66340665222888927928543807467, −4.02125708518221809874551826142, −2.57065998078734261502241065919, −1.46686180927517872610812203062, −0.51341977881179666706024373274, 2.04019124415443461737073634313, 2.79454867483433052481827070279, 3.51043025878399467315072527523, 4.75188689505607587174472311312, 5.40827160720220275282425380912, 6.56074153995635840490600298090, 7.16675343031098727940399313011, 7.938088979886346415748746585234, 8.660828175114258050157836438852, 9.778740929220704151201737279124

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