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Properties

Label 2-231-11.4-c1-0-3
Degree $2$
Conductor $231$
Sign $-0.999 - 0.0439i$
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  + (−2.11 − 1.53i)2-s + (−0.309 + 0.951i)3-s + (1.5 + 4.61i)4-s + (−1.30 + 0.951i)5-s + (2.11 − 1.53i)6-s + (0.309 + 0.951i)7-s + (2.30 − 7.10i)8-s + (−0.809 − 0.587i)9-s + 4.23·10-s + (−2.54 − 2.12i)11-s − 4.85·12-s + (−4.42 − 3.21i)13-s + (0.809 − 2.48i)14-s + (−0.499 − 1.53i)15-s + (−7.97 + 5.79i)16-s + (2.42 − 1.76i)17-s + ⋯
L(s)  = 1  + (−1.49 − 1.08i)2-s + (−0.178 + 0.549i)3-s + (0.750 + 2.30i)4-s + (−0.585 + 0.425i)5-s + (0.864 − 0.628i)6-s + (0.116 + 0.359i)7-s + (0.816 − 2.51i)8-s + (−0.269 − 0.195i)9-s + 1.33·10-s + (−0.767 − 0.641i)11-s − 1.40·12-s + (−1.22 − 0.892i)13-s + (0.216 − 0.665i)14-s + (−0.129 − 0.397i)15-s + (−1.99 + 1.44i)16-s + (0.588 − 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.999 - 0.0439i$
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: \((2,\ 231,\ (\ :1/2),\ -0.999 - 0.0439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00202661 + 0.0921638i\)
\(L(\frac12)\) \(\approx\) \(0.00202661 + 0.0921638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 2.12i)T \)
good2 \( 1 + (2.11 + 1.53i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.42 + 3.21i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.42 + 1.76i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.23 + 3.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + (-0.927 - 2.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.190 + 0.138i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.61 + 8.05i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.954 - 2.93i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (-3.11 + 9.59i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.30 + 2.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.89 - 12.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.42 + 3.94i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 3.70T + 67T^{2} \)
71 \( 1 + (9.89 - 7.19i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.19 + 3.66i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.54 + 4.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.38 - 1.73i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.67T + 89T^{2} \)
97 \( 1 + (-13.2 - 9.64i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.57204646252305713031651721673, −10.50405368132859352427832258771, −10.04772605988212059908424381759, −8.973432883554362188689702707892, −7.993898932392597326638770060303, −7.27785477966797319823951118705, −5.31142028221977860744088455203, −3.50248675518077223148018888677, −2.56220895859971984361298126914, −0.11886784829226630252888362140, 1.80033546662995535019263700637, 4.67417751463471007293925104374, 5.95177059252378348657252491087, 7.06914598955019881708500646147, 7.82987854008509418057086213217, 8.331843907894732227599571819918, 9.804584641946643686139246666044, 10.21096154541276022104112759940, 11.67285586420006387315683342130, 12.40359623213200877222693095870

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