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Properties

Label 1-2028-2028.407-r0-0-0
Degree $1$
Conductor $2028$
Sign $0.272 - 0.962i$
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  + (0.568 + 0.822i)5-s + (−0.200 − 0.979i)7-s + (−0.278 − 0.960i)11-s + (0.200 + 0.979i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.354 + 0.935i)25-s + (−0.692 − 0.721i)29-s + (−0.354 − 0.935i)31-s + (0.692 − 0.721i)35-s + (0.632 − 0.774i)37-s + (0.799 − 0.600i)41-s + (0.632 + 0.774i)43-s + (−0.885 − 0.464i)47-s + (−0.919 + 0.391i)49-s + ⋯
L(s)  = 1  + (0.568 + 0.822i)5-s + (−0.200 − 0.979i)7-s + (−0.278 − 0.960i)11-s + (0.200 + 0.979i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.354 + 0.935i)25-s + (−0.692 − 0.721i)29-s + (−0.354 − 0.935i)31-s + (0.692 − 0.721i)35-s + (0.632 − 0.774i)37-s + (0.799 − 0.600i)41-s + (0.632 + 0.774i)43-s + (−0.885 − 0.464i)47-s + (−0.919 + 0.391i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $0.272 - 0.962i$
Analytic conductor: \(9.41799\)
Root analytic conductor: \(9.41799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2028,\ (0:\ ),\ 0.272 - 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062946866 - 0.8039176574i\)
\(L(\frac12)\) \(\approx\) \(1.062946866 - 0.8039176574i\)
\(L(1)\) \(\approx\) \(1.043392088 - 0.1238924229i\)
\(L(1)\) \(\approx\) \(1.043392088 - 0.1238924229i\)

Euler product

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

−20.097658873870620740143308143850, −19.47680301581435686768483751112, −18.4096560224005059722814493851, −17.92032074410281182647204271135, −17.29469662112510520873197074977, −16.29673504386400215816518432138, −15.81966622179460822458715854716, −15.02063415341733904033306999360, −14.24299239489950504329670985062, −13.251075687205478429279590463958, −12.79624417137357638723709986015, −12.06451639344017267130292398631, −11.3708099757485283513729863386, −10.23859130485721435786113126000, −9.44739328699949968629317989553, −9.09735570043720998032613280084, −8.186560951886539971785717291406, −7.250281844607962929557514464718, −6.40010140768241435875309075083, −5.3545281641230887228926648380, −5.08276355132215230778841278666, −4.0411097109187094913602605318, −2.7553905713037586223926398091, −2.14951092926188017175647690597, −1.13882246619073105826642087847, 0.46684522974792958499891755179, 1.77642926345669031030408062694, 2.59786724392897551361963114196, 3.715020473097251139314372792178, 4.07687626640266623906792494122, 5.60201121705392457646480469701, 6.08013179304820055013625979210, 6.840045731919091997748820138812, 7.78762512788937289668302058266, 8.357739275814187598732316642527, 9.609569633581404621761128463, 10.13557195574415411237787258635, 10.89972071438366726305438372109, 11.30458462901137355945449914704, 12.69517937427704427151484051350, 13.14183158326551293449379011098, 14.07663688369061727025038129592, 14.45840455629531802198814017872, 15.270154820858313690028609881623, 16.43435184845661250328396299125, 16.73818431958374920644018211339, 17.61186868291133838309487211319, 18.35261292141916077450460137362, 19.083494386415689144880770343887, 19.56682390749802623991037381948

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