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Properties

Label 2-231-21.20-c1-0-4
Degree $2$
Conductor $231$
Sign $-0.949 + 0.314i$
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  + 2.33i·2-s + (1.41 + 1.00i)3-s − 3.44·4-s − 4.22·5-s + (−2.33 + 3.29i)6-s + (2.13 + 1.56i)7-s − 3.36i·8-s + (0.990 + 2.83i)9-s − 9.84i·10-s − i·11-s + (−4.86 − 3.45i)12-s − 1.39i·13-s + (−3.65 + 4.97i)14-s + (−5.96 − 4.23i)15-s + 0.965·16-s + 1.76·17-s + ⋯
L(s)  = 1  + 1.64i·2-s + (0.815 + 0.578i)3-s − 1.72·4-s − 1.88·5-s + (−0.954 + 1.34i)6-s + (0.805 + 0.592i)7-s − 1.18i·8-s + (0.330 + 0.943i)9-s − 3.11i·10-s − 0.301i·11-s + (−1.40 − 0.996i)12-s − 0.387i·13-s + (−0.977 + 1.32i)14-s + (−1.53 − 1.09i)15-s + 0.241·16-s + 0.427·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.949 + 0.314i$
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: \((2,\ 231,\ (\ :1/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174994 - 1.08558i\)
\(L(\frac12)\) \(\approx\) \(0.174994 - 1.08558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.41 - 1.00i)T \)
7 \( 1 + (-2.13 - 1.56i)T \)
11 \( 1 + iT \)
good2 \( 1 - 2.33iT - 2T^{2} \)
5 \( 1 + 4.22T + 5T^{2} \)
13 \( 1 + 1.39iT - 13T^{2} \)
17 \( 1 - 1.76T + 17T^{2} \)
19 \( 1 - 0.456iT - 19T^{2} \)
23 \( 1 - 3.57iT - 23T^{2} \)
29 \( 1 - 6.15iT - 29T^{2} \)
31 \( 1 - 2.68iT - 31T^{2} \)
37 \( 1 - 3.07T + 37T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 - 0.712T + 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 - 9.93T + 59T^{2} \)
61 \( 1 + 0.721iT - 61T^{2} \)
67 \( 1 - 2.01T + 67T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 + 7.85iT - 73T^{2} \)
79 \( 1 + 7.33T + 79T^{2} \)
83 \( 1 + 9.00T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 6.09iT - 97T^{2} \)
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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

−12.97271128385903933078292077539, −11.75361095240227122857436269790, −10.81698047573118373903892145926, −9.184620761962849047484687844039, −8.332102268699683013842144344616, −7.929714420119420329560397670855, −7.12487827544484891687786475825, −5.38233986880280041874924231398, −4.52097171081790343634153709058, −3.38192176254255579428431910134, 0.910333036751421063200703692956, 2.60254673672194809186009514159, 3.93134877040380628657171719140, 4.37168732233235165559007067798, 7.09777476763299521771268287591, 7.944350792664345968624805193626, 8.696673624551599258428785812218, 9.929251806223880391208536787141, 11.08869669153625771062463763250, 11.67621533519945793493418960305

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