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Properties

Label 1-845-845.42-r1-0-0
Degree $1$
Conductor $845$
Sign $-0.758 + 0.652i$
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  + (−0.0804 − 0.996i)2-s + (0.534 + 0.845i)3-s + (−0.987 + 0.160i)4-s + (0.799 − 0.600i)6-s + (−0.721 + 0.692i)7-s + (0.239 + 0.970i)8-s + (−0.428 + 0.903i)9-s + (0.428 + 0.903i)11-s + (−0.663 − 0.748i)12-s + (0.748 + 0.663i)14-s + (0.948 − 0.316i)16-s + (−0.721 + 0.692i)17-s + (0.935 + 0.354i)18-s + (0.5 + 0.866i)19-s + (−0.970 − 0.239i)21-s + (0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + (−0.0804 − 0.996i)2-s + (0.534 + 0.845i)3-s + (−0.987 + 0.160i)4-s + (0.799 − 0.600i)6-s + (−0.721 + 0.692i)7-s + (0.239 + 0.970i)8-s + (−0.428 + 0.903i)9-s + (0.428 + 0.903i)11-s + (−0.663 − 0.748i)12-s + (0.748 + 0.663i)14-s + (0.948 − 0.316i)16-s + (−0.721 + 0.692i)17-s + (0.935 + 0.354i)18-s + (0.5 + 0.866i)19-s + (−0.970 − 0.239i)21-s + (0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.758 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.758 + 0.652i$
Analytic conductor: \(90.8078\)
Root analytic conductor: \(90.8078\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 845,\ (1:\ ),\ -0.758 + 0.652i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5077545391 + 1.368652607i\)
\(L(\frac12)\) \(\approx\) \(0.5077545391 + 1.368652607i\)
\(L(1)\) \(\approx\) \(0.9752311277 + 0.2122041398i\)
\(L(1)\) \(\approx\) \(0.9752311277 + 0.2122041398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0804 - 0.996i)T \)
3 \( 1 + (0.534 + 0.845i)T \)
7 \( 1 + (-0.721 + 0.692i)T \)
11 \( 1 + (0.428 + 0.903i)T \)
17 \( 1 + (-0.721 + 0.692i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.996 - 0.0804i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.391 - 0.919i)T \)
41 \( 1 + (-0.845 + 0.534i)T \)
43 \( 1 + (0.391 + 0.919i)T \)
47 \( 1 + (0.935 - 0.354i)T \)
53 \( 1 + (0.239 + 0.970i)T \)
59 \( 1 + (-0.948 - 0.316i)T \)
61 \( 1 + (0.278 + 0.960i)T \)
67 \( 1 + (-0.160 + 0.987i)T \)
71 \( 1 + (-0.0402 - 0.999i)T \)
73 \( 1 + (-0.822 - 0.568i)T \)
79 \( 1 + (0.354 + 0.935i)T \)
83 \( 1 + (-0.464 - 0.885i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.979 - 0.200i)T \)
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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

−21.96192022726601863280101139131, −20.580901530835879025423271922122, −19.80575408467200631785793645160, −19.03504487943197264057589607279, −18.49675123170757553765145407339, −17.44869794224588682777736911219, −16.91641874830986390457287649555, −15.94205519817818199592223076102, −15.221897148416665863063510955923, −14.139266184700625421450487999637, −13.60681708913379776811765496243, −13.148432559251591858240748839377, −12.05686740882088754676531400504, −10.9207563911376575206657320783, −9.670857079489829963332379238882, −8.98422397906625192994577378623, −8.26321839986803800890231806114, −7.17280776511988884810460525703, −6.76413382725210176202525572301, −5.96339459841540492329910324208, −4.7062600319401076312496535803, −3.611814111779999912834982888623, −2.728436254088851283786496691887, −0.96691961716607743683650687050, −0.37450931291238257177167535539, 1.461907771983505082275881421, 2.504396079057476558460106542313, 3.2784426825989272980119113552, 4.16415791488162889399301134684, 4.9841153513005680917889661592, 6.049255901656955207542563534470, 7.48414848059364648503608639631, 8.660299617985514823162529623559, 9.13145469533066903958728581289, 9.95532428056286751987535519323, 10.538223042966754750265882560590, 11.61610559503618416055966998706, 12.40854919486894233642980104347, 13.17899292014163363604943335968, 14.13323355976816842764795445373, 14.924217822427063675686167019732, 15.67112459207169543387540282747, 16.6595342636472451985688416461, 17.54493703569493714911979957827, 18.427085030124724650082206028618, 19.49032779732918854885847523279, 19.71725933552745478190910614945, 20.63217761933350365250473941855, 21.45282684218936043160724370324, 21.956184543478742179218047399162

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