↑

Properties

Label 1-373-373.29-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.259 - 0.965i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.0843 − 0.996i)2-s + (−0.315 − 0.948i)3-s + (−0.985 − 0.168i)4-s + (0.990 + 0.134i)5-s + (−0.972 + 0.234i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (−0.801 + 0.598i)9-s + (0.217 − 0.975i)10-s + (0.963 + 0.266i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.890 − 0.455i)14-s + (−0.184 − 0.982i)15-s + (0.943 + 0.331i)16-s + (0.347 + 0.937i)17-s + ⋯
L(s)  = 1  + (0.0843 − 0.996i)2-s + (−0.315 − 0.948i)3-s + (−0.985 − 0.168i)4-s + (0.990 + 0.134i)5-s + (−0.972 + 0.234i)6-s + (0.528 + 0.848i)7-s + (−0.250 + 0.968i)8-s + (−0.801 + 0.598i)9-s + (0.217 − 0.975i)10-s + (0.963 + 0.266i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.890 − 0.455i)14-s + (−0.184 − 0.982i)15-s + (0.943 + 0.331i)16-s + (0.347 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.259 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8541555891 - 1.114007365i\)
\(L(\frac12)\) \(\approx\) \(0.8541555891 - 1.114007365i\)
\(L(1)\) \(\approx\) \(0.9034994247 - 0.7215138251i\)
\(L(1)\) \(\approx\) \(0.9034994247 - 0.7215138251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.0843 - 0.996i)T \)
3 \( 1 + (-0.315 - 0.948i)T \)
5 \( 1 + (0.990 + 0.134i)T \)
7 \( 1 + (0.528 + 0.848i)T \)
11 \( 1 + (0.963 + 0.266i)T \)
13 \( 1 + (-0.250 - 0.968i)T \)
17 \( 1 + (0.347 + 0.937i)T \)
19 \( 1 + (-0.758 - 0.651i)T \)
23 \( 1 + (0.979 - 0.201i)T \)
29 \( 1 + (0.409 - 0.912i)T \)
31 \( 1 + (0.151 - 0.988i)T \)
37 \( 1 + (0.857 - 0.514i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (-0.985 - 0.168i)T \)
47 \( 1 + (-0.839 + 0.543i)T \)
53 \( 1 + (-0.801 - 0.598i)T \)
59 \( 1 + (0.585 + 0.810i)T \)
61 \( 1 + (-0.315 - 0.948i)T \)
67 \( 1 + (-0.612 + 0.790i)T \)
71 \( 1 + (0.638 + 0.769i)T \)
73 \( 1 + (-0.117 + 0.993i)T \)
79 \( 1 + (0.217 - 0.975i)T \)
83 \( 1 + (-0.801 - 0.598i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.347 + 0.937i)T \)
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

−25.05635550397187759459773245752, −23.94543341431388469037010882492, −23.167699776712699824374921975088, −22.308159633860291304735903345123, −21.42831609792196314581886161524, −20.95576874501357982866721324537, −19.63956621941948944959419997539, −18.28452420527303904506642571568, −17.411741270463732972889629395484, −16.66454561886947669897758275686, −16.45652172376077188387526076566, −14.91631366765873094831883287847, −14.26310724793798970459222432607, −13.76079990089365294463067184745, −12.348987187091553254215155711968, −11.12502262760627730743299427898, −10.04493108597324019272523919, −9.299012359840178761068447438024, −8.53195090100492609271776766983, −6.99165381917055162342613180871, −6.28969775203443570682381496789, −5.08900005198834547514173302202, −4.513641471405941245956540964, −3.35883721783154457490311680463, −1.24098980084386131746512540925, 1.1327407255259777924596047640, 2.09209165668942167913537810441, 2.86170996686798655966455913899, 4.63612917576009306858810111802, 5.67070883681508518308568779208, 6.386355102318741931045649110320, 7.9908131155697547424187627089, 8.86325631059909820172505810086, 9.87357477575047589810926211933, 10.98334583506496969906349814950, 11.710504396652973256880267076682, 12.830541461954396377630827149088, 13.08413701183948776699819397105, 14.453569423125418271458135781713, 14.89132210420782558747136740361, 17.15926201019501433187110624190, 17.39595828845530192174214639854, 18.2651937126355439299303945678, 19.08155930424039756590845602838, 19.8290615200559530762562920066, 20.97476542021560876122991717369, 21.75462052035276628047348160250, 22.4439318251589494926451731372, 23.27109612326960805203005188362, 24.46102557628942209390671221898

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