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Properties

Label 1-2028-2028.119-r1-0-0
Degree $1$
Conductor $2028$
Sign $0.272 + 0.962i$
Analytic cond. $217.938$
Root an. cond. $217.938$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.822 + 0.568i)5-s + (0.979 + 0.200i)7-s + (0.960 + 0.278i)11-s + (−0.200 + 0.979i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)23-s + (0.354 + 0.935i)25-s + (−0.692 + 0.721i)29-s + (−0.935 − 0.354i)31-s + (0.692 + 0.721i)35-s + (0.774 − 0.632i)37-s + (−0.600 + 0.799i)41-s + (−0.632 + 0.774i)43-s + (0.464 + 0.885i)47-s + (0.919 + 0.391i)49-s + ⋯
L(s)  = 1  + (0.822 + 0.568i)5-s + (0.979 + 0.200i)7-s + (0.960 + 0.278i)11-s + (−0.200 + 0.979i)17-s + (0.866 − 0.5i)19-s + (0.5 − 0.866i)23-s + (0.354 + 0.935i)25-s + (−0.692 + 0.721i)29-s + (−0.935 − 0.354i)31-s + (0.692 + 0.721i)35-s + (0.774 − 0.632i)37-s + (−0.600 + 0.799i)41-s + (−0.632 + 0.774i)43-s + (0.464 + 0.885i)47-s + (0.919 + 0.391i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(217.938\)
Root analytic conductor: \(217.938\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (1:\ ),\ 0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.887004121 + 2.183470937i\)
\(L(\frac12)\) \(\approx\) \(2.887004121 + 2.183470937i\)
\(L(1)\) \(\approx\) \(1.509307751 + 0.3701058557i\)
\(L(1)\) \(\approx\) \(1.509307751 + 0.3701058557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (0.822 + 0.568i)T \)
7 \( 1 + (0.979 + 0.200i)T \)
11 \( 1 + (0.960 + 0.278i)T \)
17 \( 1 + (-0.200 + 0.979i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.692 + 0.721i)T \)
31 \( 1 + (-0.935 - 0.354i)T \)
37 \( 1 + (0.774 - 0.632i)T \)
41 \( 1 + (-0.600 + 0.799i)T \)
43 \( 1 + (-0.632 + 0.774i)T \)
47 \( 1 + (0.464 + 0.885i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (0.0804 - 0.996i)T \)
61 \( 1 + (0.948 + 0.316i)T \)
67 \( 1 + (-0.999 - 0.0402i)T \)
71 \( 1 + (0.391 + 0.919i)T \)
73 \( 1 + (-0.239 - 0.970i)T \)
79 \( 1 + (-0.885 + 0.464i)T \)
83 \( 1 + (0.992 - 0.120i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.903 - 0.428i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.81312546407869381997722310318, −18.66673460209686550011590077154, −18.10488814550167070798054789912, −17.324348772363990077143056269771, −16.83362565473833498576303735105, −16.13384124106007863736572520053, −15.06565727396827120860011088616, −14.39541563858563313947084335185, −13.64326277233697622430322591484, −13.28561845788662400852944224769, −11.88422651902442213453650192262, −11.73843949507369092282969258349, −10.69098894783771911813469410929, −9.78131018072253441480153268034, −9.145462365741740532574288302386, −8.49737716199684687517797079358, −7.496565267176859238268815097511, −6.80784161152537617793199815606, −5.60673743156035418636620440566, −5.27336239955141782226582355937, −4.28042862895853653624229496241, −3.40244611343902718905752282768, −2.14369134608345151448835574448, −1.44334767735414265900943223898, −0.640983616487057561586714447566, 1.112630413069548306859160290520, 1.79337714401089645289877788265, 2.64838916796183231459575426852, 3.69509816366370214426858636544, 4.63448222594423888114814267861, 5.4631146763063047912606892281, 6.24887613410733364721696147405, 7.01366055175153875640931030552, 7.80317829880087256291859024934, 8.86915192272394074501433503682, 9.34086856296115409932162187221, 10.31106064760108603883951775473, 11.06998877345839880477987890358, 11.56579346469767519770324186657, 12.63315279234841517625478965672, 13.29302159851484468857472713542, 14.30247104543623170355057469767, 14.654756400174194007194822660743, 15.20598842006056860965506595583, 16.47765385573988477674509573976, 17.07101275568349711322746159431, 17.81708409106377408963432023218, 18.25327655704916211547711908657, 19.05008060001009237234243532717, 20.032774880946983100414248733842

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