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Properties

Label 1-231-231.179-r1-0-0
Degree $1$
Conductor $231$
Sign $0.325 - 0.945i$
Analytic cond. $24.8243$
Root an. cond. $24.8243$
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  + (−0.669 − 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.309 + 0.951i)20-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.913 + 0.406i)26-s + (0.809 + 0.587i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)32-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.104 + 0.994i)4-s + (0.978 + 0.207i)5-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)10-s + (0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.309 + 0.951i)20-s + (0.5 − 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.913 + 0.406i)26-s + (0.809 + 0.587i)29-s + (−0.978 + 0.207i)31-s + (0.5 + 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(24.8243\)
Root analytic conductor: \(24.8243\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 231,\ (1:\ ),\ 0.325 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.266041568 - 0.9029466085i\)
\(L(\frac12)\) \(\approx\) \(1.266041568 - 0.9029466085i\)
\(L(1)\) \(\approx\) \(0.9082509328 - 0.3302203554i\)
\(L(1)\) \(\approx\) \(0.9082509328 - 0.3302203554i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.104 - 0.994i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−26.19515780475897881411104066248, −25.323627478340251019078010765594, −24.7279592863545768564959273712, −23.72961743681466814340830705560, −22.790070056350538068946516233392, −21.5957206522398036938329412488, −20.66007622093892943686355142287, −19.55684573848943407905403261620, −18.48887313167043684422884751129, −17.82478299012513069969657118707, −16.80328862284347862239322095883, −16.1750615403140504574756407935, −14.93434039791351959646893047226, −13.977462214488344795732238084440, −13.25740403435159547048065627138, −11.63581380392804741711356608769, −10.48258098980726931703803150418, −9.46253451625132295636414572481, −8.85221857838490243642639403606, −7.513863085711434584684792267756, −6.434875319711753605489113197420, −5.59509458251181418244442164427, −4.38970049617105100534396182635, −2.2894554828538254556728265333, −1.11592653010928588474292549131, 0.7581861701880639785902802431, 2.13771912277128420943378367403, 3.10386160856633212746194214255, 4.61411657096292324825950246926, 6.083675348587776035947268632, 7.255348874450054351005490794181, 8.59350597836953322730027983731, 9.3356646731836891424395909332, 10.58315369588288568523956602161, 10.95497204914964741795481111367, 12.61954392108792290597094278247, 13.11566846209800726714905622037, 14.32313766797862205418808088583, 15.640867742480233423965409972294, 16.84538090777293315879492356259, 17.72053911319182935612733123377, 18.22100820024406045757577183595, 19.44484393588738487210749532961, 20.27333792857995462082671064374, 21.233978169128501962404321479314, 21.968234649672971489549195003014, 22.77958151608858249701771387710, 24.291288901062948508809186624990, 25.358474207157955602272396954151, 25.952203081845545956498109957346

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