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Properties

Label 2-2028-13.9-c1-0-18
Degree $2$
Conductor $2028$
Sign $0.477 + 0.878i$
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 + 4·5-s + (−1 − 1.73i)7-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + (2 − 3.46i)15-s + (−1 − 1.73i)17-s + (−1 − 1.73i)19-s − 1.99·21-s + 11·25-s − 0.999·27-s + (3 − 5.19i)29-s + 10·31-s + (1.99 + 3.46i)33-s + (−4 − 6.92i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 1.78·5-s + (−0.377 − 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + (0.516 − 0.894i)15-s + (−0.242 − 0.420i)17-s + (−0.229 − 0.397i)19-s − 0.436·21-s + 2.20·25-s − 0.192·27-s + (0.557 − 0.964i)29-s + 1.79·31-s + (0.348 + 0.603i)33-s + (−0.676 − 1.17i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(16.1936\)
Root analytic conductor: \(4.02413\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.517362826\)
\(L(\frac12)\) \(\approx\) \(2.517362826\)
\(L(\frac{3}{2})\) 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 \( 1 - 4T + 5T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8 - 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.194451947185205718992993429455, −8.247271430695410543713019468899, −7.25379609071233021423264444967, −6.69990672553677441919711384000, −5.95375484370781479810276284044, −5.09379484238728948272175329033, −4.14911773908520590507056228902, −2.58057376025299250781887517720, −2.26578326223085285246576672217, −0.907334204027431898169670422948, 1.39109178441691123336395916734, 2.65948732034728976965534537781, 3.03230261844377489867529308614, 4.59077883920367523306090668346, 5.35275369310859829874920168926, 6.21060062807430005468987044002, 6.42101021653179309198283013470, 8.084268113810508752239336887357, 8.567551169945668563040538936329, 9.447079898077029011759311171300

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