↑

Properties

Label 1-373-373.80-r1-0-0
Degree $1$
Conductor $373$
Sign $0.326 - 0.945i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.975 + 0.217i)2-s + (−0.409 − 0.912i)3-s + (0.905 − 0.425i)4-s + (−0.830 − 0.557i)5-s + (0.598 + 0.801i)6-s + (0.874 + 0.485i)7-s + (−0.790 + 0.612i)8-s + (−0.664 + 0.747i)9-s + (0.931 + 0.363i)10-s + (0.925 − 0.378i)11-s + (−0.758 − 0.651i)12-s + (0.612 − 0.790i)13-s + (−0.959 − 0.283i)14-s + (−0.168 + 0.985i)15-s + (0.638 − 0.769i)16-s + (−0.820 − 0.571i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.217i)2-s + (−0.409 − 0.912i)3-s + (0.905 − 0.425i)4-s + (−0.830 − 0.557i)5-s + (0.598 + 0.801i)6-s + (0.874 + 0.485i)7-s + (−0.790 + 0.612i)8-s + (−0.664 + 0.747i)9-s + (0.931 + 0.363i)10-s + (0.925 − 0.378i)11-s + (−0.758 − 0.651i)12-s + (0.612 − 0.790i)13-s + (−0.959 − 0.283i)14-s + (−0.168 + 0.985i)15-s + (0.638 − 0.769i)16-s + (−0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8190669131 - 0.5835257798i\)
\(L(\frac12)\) \(\approx\) \(0.8190669131 - 0.5835257798i\)
\(L(1)\) \(\approx\) \(0.6264060549 - 0.1991206970i\)
\(L(1)\) \(\approx\) \(0.6264060549 - 0.1991206970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.975 + 0.217i)T \)
3 \( 1 + (-0.409 - 0.912i)T \)
5 \( 1 + (-0.830 - 0.557i)T \)
7 \( 1 + (0.874 + 0.485i)T \)
11 \( 1 + (0.925 - 0.378i)T \)
13 \( 1 + (0.612 - 0.790i)T \)
17 \( 1 + (-0.820 - 0.571i)T \)
19 \( 1 + (-0.937 - 0.347i)T \)
23 \( 1 + (0.101 + 0.994i)T \)
29 \( 1 + (0.890 + 0.455i)T \)
31 \( 1 + (0.758 - 0.651i)T \)
37 \( 1 + (0.713 + 0.701i)T \)
41 \( 1 + (0.918 + 0.394i)T \)
43 \( 1 + (0.425 + 0.905i)T \)
47 \( 1 + (-0.234 + 0.972i)T \)
53 \( 1 + (0.747 - 0.664i)T \)
59 \( 1 + (0.839 - 0.543i)T \)
61 \( 1 + (-0.912 + 0.409i)T \)
67 \( 1 + (0.897 - 0.440i)T \)
71 \( 1 + (-0.0843 - 0.996i)T \)
73 \( 1 + (-0.315 + 0.948i)T \)
79 \( 1 + (0.363 - 0.931i)T \)
83 \( 1 + (-0.664 - 0.747i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.571 + 0.820i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.64412247477755446341957649269, −23.608512001661121988089306210431, −22.83674268818624748124410585622, −21.68165148614785593863892206072, −21.03511034679990419267888059521, −20.070117123464255848917805448264, −19.41329581794555909102113851017, −18.292424566871244152532340595355, −17.41951566573570128308470781629, −16.75528874270027912379324995544, −15.82580609862891902509370457401, −14.99233205576215489032468799641, −14.26009894984352449787644785246, −12.29231584358223228305968777462, −11.54776155215729462425413568712, −10.83717957464791895335427262036, −10.278728205590768933499985207936, −8.88582318172304221537816657760, −8.345699490920943095592149241890, −6.96050308138655989264705377579, −6.28624820308820505553248093774, −4.27449355715658446415142206030, −3.96730492639913661840641242515, −2.32140729848723848827706466927, −0.80554740595551284022119643698, 0.64624520873894715180771870704, 1.4217591761639262689652981665, 2.77545682875003327957527360379, 4.624976108418658003192627167036, 5.81375379506133188675803747076, 6.71509410993101842071245444082, 7.872579830430833209677550583553, 8.36102947914842839889077363567, 9.21272364722465219256062162880, 10.99899317212600825351949502954, 11.3848754551394885955820865789, 12.17392724425642422811463563348, 13.31105224186760674824609993809, 14.6253617467411326745114442259, 15.54202768081546891809589449558, 16.41042281177806181618382190877, 17.45614663148918614856714740563, 17.86068813462543560965700588416, 18.94819745645392327868385756826, 19.59236655472396038243631060347, 20.32104749806495658873762035155, 21.44416048033416638100947578921, 22.811655004813032416999621955017, 23.68052734939303396064274867670, 24.43415376612224111197317710974

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