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Properties

Label 2-231-21.20-c1-0-24
Degree $2$
Conductor $231$
Sign $-0.981 + 0.189i$
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  − 0.776i·2-s + (−0.233 − 1.71i)3-s + 1.39·4-s − 3.58·5-s + (−1.33 + 0.181i)6-s + (−2.64 + 0.145i)7-s − 2.63i·8-s + (−2.89 + 0.803i)9-s + 2.78i·10-s − i·11-s + (−0.326 − 2.39i)12-s − 4.05i·13-s + (0.112 + 2.05i)14-s + (0.838 + 6.15i)15-s + 0.744·16-s + 4.26·17-s + ⋯
L(s)  = 1  − 0.549i·2-s + (−0.135 − 0.990i)3-s + 0.698·4-s − 1.60·5-s + (−0.544 + 0.0741i)6-s + (−0.998 + 0.0548i)7-s − 0.932i·8-s + (−0.963 + 0.267i)9-s + 0.880i·10-s − 0.301i·11-s + (−0.0943 − 0.691i)12-s − 1.12i·13-s + (0.0301 + 0.548i)14-s + (0.216 + 1.58i)15-s + 0.186·16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.981 + 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0744995 - 0.780382i\)
\(L(\frac12)\) \(\approx\) \(0.0744995 - 0.780382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.233 + 1.71i)T \)
7 \( 1 + (2.64 - 0.145i)T \)
11 \( 1 + iT \)
good2 \( 1 + 0.776iT - 2T^{2} \)
5 \( 1 + 3.58T + 5T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 - 4.26T + 17T^{2} \)
19 \( 1 - 2.75iT - 19T^{2} \)
23 \( 1 + 7.51iT - 23T^{2} \)
29 \( 1 - 5.24iT - 29T^{2} \)
31 \( 1 + 6.53iT - 31T^{2} \)
37 \( 1 + 2.71T + 37T^{2} \)
41 \( 1 + 4.19T + 41T^{2} \)
43 \( 1 - 0.621T + 43T^{2} \)
47 \( 1 - 7.15T + 47T^{2} \)
53 \( 1 + 1.37iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 0.588iT - 61T^{2} \)
67 \( 1 + 7.60T + 67T^{2} \)
71 \( 1 + 8.94iT - 71T^{2} \)
73 \( 1 - 9.68iT - 73T^{2} \)
79 \( 1 - 5.10T + 79T^{2} \)
83 \( 1 + 1.24T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−12.04153246050821652152195433377, −10.98768168774063024054333867284, −10.20722021947215565974122221256, −8.485171532210251820205610415001, −7.65770112609731221026710705881, −6.86532481587749725423786630873, −5.76760258575707631685483243778, −3.66153220416063348532611766060, −2.83011859225626038983218841503, −0.63796075650246751829770498832, 3.13413899442498241730562281053, 4.07263420416351534458108275003, 5.43021640155383660365449994478, 6.74766757178020583988763159606, 7.53132542760519088164285074622, 8.672360045776947356299776208992, 9.749888909902505007032734285151, 10.84669909841455243130522053805, 11.75868029967700008793546008595, 12.11868295485173983779009397668

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