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Properties

Label 1-2028-2028.815-r0-0-0
Degree $1$
Conductor $2028$
Sign $0.0711 - 0.997i$
Analytic cond. $9.41799$
Root an. cond. $9.41799$
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  + (−0.120 − 0.992i)5-s + (0.845 − 0.534i)7-s + (0.987 − 0.160i)11-s + (0.845 − 0.534i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.970 + 0.239i)25-s + (0.632 − 0.774i)29-s + (0.970 + 0.239i)31-s + (−0.632 − 0.774i)35-s + (0.278 + 0.960i)37-s + (0.996 + 0.0804i)41-s + (−0.278 + 0.960i)43-s + (−0.748 − 0.663i)47-s + (0.428 − 0.903i)49-s + ⋯
L(s)  = 1  + (−0.120 − 0.992i)5-s + (0.845 − 0.534i)7-s + (0.987 − 0.160i)11-s + (0.845 − 0.534i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.970 + 0.239i)25-s + (0.632 − 0.774i)29-s + (0.970 + 0.239i)31-s + (−0.632 − 0.774i)35-s + (0.278 + 0.960i)37-s + (0.996 + 0.0804i)41-s + (−0.278 + 0.960i)43-s + (−0.748 − 0.663i)47-s + (0.428 − 0.903i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0711 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.0711 - 0.997i$
Analytic conductor: \(9.41799\)
Root analytic conductor: \(9.41799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (0:\ ),\ 0.0711 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485569560 - 1.383309549i\)
\(L(\frac12)\) \(\approx\) \(1.485569560 - 1.383309549i\)
\(L(1)\) \(\approx\) \(1.214456017 - 0.4431107534i\)
\(L(1)\) \(\approx\) \(1.214456017 - 0.4431107534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.120 - 0.992i)T \)
7 \( 1 + (0.845 - 0.534i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
17 \( 1 + (0.845 - 0.534i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.632 - 0.774i)T \)
31 \( 1 + (0.970 + 0.239i)T \)
37 \( 1 + (0.278 + 0.960i)T \)
41 \( 1 + (0.996 + 0.0804i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.0402 + 0.999i)T \)
67 \( 1 + (0.200 - 0.979i)T \)
71 \( 1 + (0.428 + 0.903i)T \)
73 \( 1 + (-0.354 + 0.935i)T \)
79 \( 1 + (0.748 + 0.663i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.799 + 0.600i)T \)
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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

−20.00459429637643938989989556771, −19.222692875156292449286650166225, −18.719132914907462490085782472999, −17.81466675832499157787326603740, −17.478948417675417520406654010977, −16.430824521937907524573787016005, −15.60191954943041707765741537608, −14.85054785346602595331762490843, −14.284939384844972572167544118976, −13.8877391702282271068795671824, −12.461505693347915960348749172852, −11.95633537700110365238207448350, −11.29373162229233882602044681715, −10.50044188300748115245382979759, −9.738405219808389801346750594401, −8.91846143095530548889521634833, −7.86044733557017066075613629752, −7.52163777979682047606109863953, −6.30990633182988749627745602079, −5.867127696731901977238146453154, −4.78760680061464100313471460666, −3.80925736446967466321278296914, −3.143910244237377284891344169335, −2.0207488615074246931677586695, −1.30318712455118824031017623098, 0.82174803285293836891873733748, 1.309114588649310075691137074526, 2.55758587784963160438483979166, 3.69819694206506570013816966318, 4.58860083899951075682305948819, 4.953569854833026730928693468324, 6.091999143532763384771190387594, 6.9151968617423504924924938657, 8.01417596125418738097194682474, 8.2920977088314465437795103995, 9.379938676586518308943893768, 9.89399994480778105992639104394, 11.02800692374201181048754031148, 11.757077920899690394806370724870, 12.177650493247395359403505859643, 13.24508810632404928095645575147, 13.91970126866016913767979237115, 14.482108464724457871157528793579, 15.4382539617616873265445125912, 16.27992511416272823269618696136, 16.8271590644072975142176919198, 17.49358113111529784326696878612, 18.129396121929216973470665950002, 19.20155383662644975372455944981, 19.85834263575222128343325593011

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