↑

Properties

Label 1-115-115.83-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.841 - 0.540i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (−0.540 − 0.841i)2-s + (−0.989 − 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (0.755 − 0.654i)7-s + (0.989 − 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.540 − 0.841i)12-s + (−0.755 − 0.654i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (−0.281 − 0.959i)18-s + (0.415 − 0.909i)19-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (−0.989 − 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (0.755 − 0.654i)7-s + (0.989 − 0.142i)8-s + (0.959 + 0.281i)9-s + (−0.841 − 0.540i)11-s + (0.540 − 0.841i)12-s + (−0.755 − 0.654i)13-s + (−0.959 − 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (−0.281 − 0.959i)18-s + (0.415 − 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1245927388 - 0.4241584791i\)
\(L(\frac12)\) \(\approx\) \(0.1245927388 - 0.4241584791i\)
\(L(1)\) \(\approx\) \(0.4407204012 - 0.3253211224i\)
\(L(1)\) \(\approx\) \(0.4407204012 - 0.3253211224i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.540 - 0.841i)T \)
3 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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

−29.03335472578621694766157141788, −28.71642445051833908898409497160, −27.44465032788597804393760986366, −26.951978918817053387639435365473, −25.6251303239577831942832166486, −24.41315445030987423789428909530, −23.868408544943186005287449555543, −22.725095820931627915495144093509, −21.78291001276745177065617674273, −20.46069822504001697666343223883, −18.806516297799026869765070293223, −18.08715954625058027465762627685, −17.293965993013218151305601302649, −16.164124399159658679752160177601, −15.34277200025143980953549211667, −14.24205406105480203085995479506, −12.62410613829793606650726660313, −11.39873320640814235962401129572, −10.28594909728841194871490669280, −9.17483110172890776308406013209, −7.750762311698418592478642833844, −6.6911786828537562206460035482, −5.33546730881803700726095184023, −4.68732753607526023340350768233, −1.79277172124179599602829784144, 0.57753415122684892443316521346, 2.2913287058957721223521948853, 4.18431010601114595685659613309, 5.321253540257619787841801442465, 7.184708746285196886077905878705, 8.12801245859982308651393720017, 9.79940823000616843192056939815, 10.86151227457555180755439631970, 11.40366728722199806274840203824, 12.76755625472687904019671232389, 13.57974830284986607079017599858, 15.48465033512114625837296508447, 16.82738240436635871155044332013, 17.56150646228965052673627503038, 18.31762759478130791618926960717, 19.54778618337243162842033372882, 20.64416778590386412658986591454, 21.67093499406387636530817871165, 22.49924917718230103370950008337, 23.710354045891107410274293252798, 24.609482613324852222687809093002, 26.40212751326286295657546403705, 26.96969791885296354263446342774, 28.0441324359296003015849680195, 28.82017689626791520450741021799

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