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Properties

Label 2-2028-13.4-c1-0-6
Degree $2$
Conductor $2028$
Sign $-0.454 - 0.890i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.454 - 0.890i$
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: \((2,\ 2028,\ (\ :1/2),\ -0.454 - 0.890i)\)

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

−9.448604226076406450394421122893, −8.652020451989679138839798826427, −7.54366224463653654311547917570, −7.02354203677682386979022534102, −6.41828526041226557928924594561, −5.81123908823528425677135855811, −4.27930073702111663760417087140, −3.28044049645570951462659754632, −2.75694569841777552706544696664, −1.61975033420697757987100896366, 0.52971129587738853294087149004, 1.60984616383238790659676246936, 3.22844972834352146454980873119, 3.99351048552983604914924096749, 4.75365138107859220109840666807, 5.56159385640590190717119270674, 6.38827513399849217724341860481, 7.51780347834573687569459934865, 8.345944306150608499652649160752, 9.005295381183170069522580907802

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