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Properties

Label 1-845-845.198-r1-0-0
Degree $1$
Conductor $845$
Sign $-0.0381 + 0.999i$
Analytic cond. $90.8078$
Root an. cond. $90.8078$
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  + (0.999 + 0.0402i)2-s + (0.960 + 0.278i)3-s + (0.996 + 0.0804i)4-s + (0.948 + 0.316i)6-s + (0.391 + 0.919i)7-s + (0.992 + 0.120i)8-s + (0.845 + 0.534i)9-s + (−0.845 + 0.534i)11-s + (0.935 + 0.354i)12-s + (0.354 + 0.935i)14-s + (0.987 + 0.160i)16-s + (0.391 + 0.919i)17-s + (0.822 + 0.568i)18-s + (0.5 + 0.866i)19-s + (0.120 + 0.992i)21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (0.999 + 0.0402i)2-s + (0.960 + 0.278i)3-s + (0.996 + 0.0804i)4-s + (0.948 + 0.316i)6-s + (0.391 + 0.919i)7-s + (0.992 + 0.120i)8-s + (0.845 + 0.534i)9-s + (−0.845 + 0.534i)11-s + (0.935 + 0.354i)12-s + (0.354 + 0.935i)14-s + (0.987 + 0.160i)16-s + (0.391 + 0.919i)17-s + (0.822 + 0.568i)18-s + (0.5 + 0.866i)19-s + (0.120 + 0.992i)21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.0381 + 0.999i$
Analytic conductor: \(90.8078\)
Root analytic conductor: \(90.8078\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 845,\ (1:\ ),\ -0.0381 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.526002698 + 4.702214794i\)
\(L(\frac12)\) \(\approx\) \(4.526002698 + 4.702214794i\)
\(L(1)\) \(\approx\) \(2.661913762 + 1.065967626i\)
\(L(1)\) \(\approx\) \(2.661913762 + 1.065967626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.999 + 0.0402i)T \)
3 \( 1 + (0.960 + 0.278i)T \)
7 \( 1 + (0.391 + 0.919i)T \)
11 \( 1 + (-0.845 + 0.534i)T \)
17 \( 1 + (0.391 + 0.919i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.0402 - 0.999i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (-0.979 + 0.200i)T \)
41 \( 1 + (0.278 - 0.960i)T \)
43 \( 1 + (-0.979 - 0.200i)T \)
47 \( 1 + (0.822 - 0.568i)T \)
53 \( 1 + (0.992 + 0.120i)T \)
59 \( 1 + (-0.987 + 0.160i)T \)
61 \( 1 + (0.799 - 0.600i)T \)
67 \( 1 + (0.0804 + 0.996i)T \)
71 \( 1 + (0.692 + 0.721i)T \)
73 \( 1 + (-0.464 + 0.885i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.774 + 0.632i)T \)
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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

−21.52075186572604110391915212749, −20.95153399201393103764352923427, −20.117943326422934869433973908254, −19.808972245205435638010052259928, −18.63600203225681194731378239966, −17.87785103499145607736517766460, −16.55746289037740732962528456910, −15.91130292405831926572262333173, −15.07033345072919252440109449443, −14.1733442820182010363979681267, −13.67890148917871020065221506748, −13.13861208756498192586084631019, −12.11606957642229451761747118491, −11.1717523483879693720363602434, −10.35809565770514205202121887223, −9.38429570810615946213499215033, −8.081527139977485812738855165513, −7.47788464234964385793221010348, −6.795739129915969859770278502, −5.486340666306079849324386188512, −4.639647043772069801822534456276, −3.60312751506510164331821264274, −2.94889369523490543733776474320, −1.89213902099913288257732882197, −0.790237979912505024677863645814, 1.75028631028868697864359467472, 2.28478516117163987396741999578, 3.339678791627051715296266433282, 4.16272894323825329775621586225, 5.179595672702557127363888924754, 5.8638993046513747196692254701, 7.18022531422779735185525893150, 7.98873674898759936426944743184, 8.65237785660853472622959825501, 10.0327433178880749055734601097, 10.5116591587849440694758424054, 11.86589375062544656637458600665, 12.47338869478882172489557328899, 13.269076522106973695841760885730, 14.23798835563981551225587179740, 14.722431799800982419073153623136, 15.57554114920302745716796438015, 15.96368648218723973007700985846, 17.16018553721223168069217092109, 18.41589200832943923719462526977, 19.02013624509024583826052705741, 20.10815029710248134125948923987, 20.638197273417261993408012663279, 21.387888474951373849976442633108, 21.88209231197128073743378406897

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