Properties

Label 2-231-11.4-c1-0-0
Degree $2$
Conductor $231$
Sign $0.0694 - 0.997i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.43 − 1.04i)2-s + (−0.309 + 0.951i)3-s + (0.352 + 1.08i)4-s + (−0.477 + 0.346i)5-s + (1.43 − 1.04i)6-s + (−0.309 − 0.951i)7-s + (−0.470 + 1.44i)8-s + (−0.809 − 0.587i)9-s + 1.04·10-s + (−2.19 + 2.48i)11-s − 1.14·12-s + (1.24 + 0.907i)13-s + (−0.547 + 1.68i)14-s + (−0.182 − 0.561i)15-s + (4.02 − 2.92i)16-s + (−5.78 + 4.19i)17-s + ⋯
L(s)  = 1  + (−1.01 − 0.736i)2-s + (−0.178 + 0.549i)3-s + (0.176 + 0.542i)4-s + (−0.213 + 0.155i)5-s + (0.585 − 0.425i)6-s + (−0.116 − 0.359i)7-s + (−0.166 + 0.512i)8-s + (−0.269 − 0.195i)9-s + 0.330·10-s + (−0.660 + 0.750i)11-s − 0.329·12-s + (0.346 + 0.251i)13-s + (−0.146 + 0.450i)14-s + (−0.0470 − 0.144i)15-s + (1.00 − 0.731i)16-s + (−1.40 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.0694 - 0.997i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.0694 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269479 + 0.251363i\)
\(L(\frac12)\) \(\approx\) \(0.269479 + 0.251363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (2.19 - 2.48i)T \)
good2 \( 1 + (1.43 + 1.04i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.477 - 0.346i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.24 - 0.907i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.78 - 4.19i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.91 - 5.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.76T + 23T^{2} \)
29 \( 1 + (-0.187 - 0.577i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.55 - 4.03i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.38 + 7.32i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.31 + 7.11i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + (3.15 - 9.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.69 + 4.14i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.03 + 3.18i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.72 + 1.98i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.19T + 67T^{2} \)
71 \( 1 + (-11.2 + 8.19i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.800 + 2.46i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.77 - 2.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.83 + 2.06i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 8.15T + 89T^{2} \)
97 \( 1 + (-3.88 - 2.81i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13001177740465251454884351447, −10.93312730157266975432102181645, −10.63516461486957591846270804629, −9.690226672902468395220189390860, −8.768814791899084952601452303923, −7.83451456528059982831969605312, −6.42930079050104179251546563077, −4.97330475350057508260129649338, −3.60735076242508079197097487333, −1.91699083699130553173402124136, 0.41654078233396458571610060174, 2.82068377865080185870176738581, 4.83674391518343089838784770792, 6.30671865823232876996929495131, 6.95776967610916953334819440641, 8.224179896052916456079162547117, 8.641866524481832823539934498663, 9.747449189361519817236505983257, 10.99174835135385818094064158091, 11.81352400032925352209431674289

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