↑

Properties

Label 1-89-89.31-r1-0-0
Degree $1$
Conductor $89$
Sign $0.981 + 0.192i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (−0.654 − 0.755i)2-s + (−0.599 + 0.800i)3-s + (−0.142 + 0.989i)4-s + (−0.540 − 0.841i)5-s + (0.997 − 0.0713i)6-s + (−0.977 − 0.212i)7-s + (0.841 − 0.540i)8-s + (−0.281 − 0.959i)9-s + (−0.281 + 0.959i)10-s + (−0.841 − 0.540i)11-s + (−0.707 − 0.707i)12-s + (0.800 + 0.599i)13-s + (0.479 + 0.877i)14-s + (0.997 + 0.0713i)15-s + (−0.959 − 0.281i)16-s + (0.755 + 0.654i)17-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.599 + 0.800i)3-s + (−0.142 + 0.989i)4-s + (−0.540 − 0.841i)5-s + (0.997 − 0.0713i)6-s + (−0.977 − 0.212i)7-s + (0.841 − 0.540i)8-s + (−0.281 − 0.959i)9-s + (−0.281 + 0.959i)10-s + (−0.841 − 0.540i)11-s + (−0.707 − 0.707i)12-s + (0.800 + 0.599i)13-s + (0.479 + 0.877i)14-s + (0.997 + 0.0713i)15-s + (−0.959 − 0.281i)16-s + (0.755 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.981 + 0.192i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ 0.981 + 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5768545771 + 0.05606127816i\)
\(L(\frac12)\) \(\approx\) \(0.5768545771 + 0.05606127816i\)
\(L(1)\) \(\approx\) \(0.5214595281 - 0.06852665490i\)
\(L(1)\) \(\approx\) \(0.5214595281 - 0.06852665490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.599 + 0.800i)T \)
5 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (-0.977 - 0.212i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (0.800 + 0.599i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (-0.479 + 0.877i)T \)
23 \( 1 + (0.877 + 0.479i)T \)
29 \( 1 + (0.212 - 0.977i)T \)
31 \( 1 + (0.877 - 0.479i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.800 + 0.599i)T \)
43 \( 1 + (0.212 + 0.977i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (-0.989 + 0.142i)T \)
59 \( 1 + (0.599 + 0.800i)T \)
61 \( 1 + (-0.349 + 0.936i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.540 - 0.841i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (0.281 - 0.959i)T \)
83 \( 1 + (-0.997 + 0.0713i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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

−30.09102133829842070877873085705, −28.951901687016347083352257175384, −28.16636500728491991381309605405, −27.0442754949837806674714804934, −25.70476987868171096173511992193, −25.32370116233412785076979394017, −23.66085683577523162228205053338, −23.163778748060464502207451884015, −22.31438690749373898212853204702, −20.118520373263391462651066792633, −18.87288322772302457452574968701, −18.523150191152293936730630697172, −17.37698011459165682766519152857, −16.0610149254199740981574557878, −15.31779781149734707419986791655, −13.79002166327032140786029312936, −12.59247040438488438566000478019, −11.0448402837147982934101423729, −10.15958433131635939367235185104, −8.41459714405815571008832821893, −7.20596070853222891370381949656, −6.519749641395952484286195650644, −5.19938946062181187911447510095, −2.746716560700225507583384148668, −0.54258572702798780319775867306, 0.84302087043118020998177952138, 3.3313423401095881391252338545, 4.30653598635787419491399171375, 6.03414203738232147440913329506, 7.97983490087090501644018951147, 9.140001648598796029209844806343, 10.169807306376957000031510528174, 11.232007215385985778946548948214, 12.3317461011557346076169665365, 13.32153306787763062624916135552, 15.57325177206997797745045009330, 16.45383673825885375056260554874, 17.01183902839979921948508099276, 18.65511293540056123029648656014, 19.54421712427901400636127936172, 20.93096764996478644183679180877, 21.269117589503390302082424337173, 22.84436476038659186813013912602, 23.55022665726925379109169676379, 25.43866598287974399910036597117, 26.47367615445960972837979328583, 27.274103409931782968847545875007, 28.51171966330570167711822691216, 28.68480201465894956133321494665, 29.95256677323465835882282576331

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