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Properties

Label 2-231-33.32-c1-0-15
Degree $2$
Conductor $231$
Sign $0.108 + 0.994i$
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  − 0.858·2-s + (1.60 − 0.639i)3-s − 1.26·4-s − 2.44i·5-s + (−1.38 + 0.549i)6-s + i·7-s + 2.80·8-s + (2.18 − 2.05i)9-s + 2.10i·10-s + (−3.19 − 0.883i)11-s + (−2.03 + 0.807i)12-s − 5.20i·13-s − 0.858i·14-s + (−1.56 − 3.93i)15-s + 0.121·16-s + 0.151·17-s + ⋯
L(s)  = 1  − 0.607·2-s + (0.929 − 0.369i)3-s − 0.631·4-s − 1.09i·5-s + (−0.564 + 0.224i)6-s + 0.377i·7-s + 0.990·8-s + (0.727 − 0.686i)9-s + 0.664i·10-s + (−0.963 − 0.266i)11-s + (−0.586 + 0.233i)12-s − 1.44i·13-s − 0.229i·14-s + (−0.404 − 1.01i)15-s + 0.0303·16-s + 0.0367·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.108 + 0.994i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.736853 - 0.660921i\)
\(L(\frac12)\) \(\approx\) \(0.736853 - 0.660921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 + 0.639i)T \)
7 \( 1 - iT \)
11 \( 1 + (3.19 + 0.883i)T \)
good2 \( 1 + 0.858T + 2T^{2} \)
5 \( 1 + 2.44iT - 5T^{2} \)
13 \( 1 + 5.20iT - 13T^{2} \)
17 \( 1 - 0.151T + 17T^{2} \)
19 \( 1 + 3.38iT - 19T^{2} \)
23 \( 1 - 5.75iT - 23T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 2.83T + 41T^{2} \)
43 \( 1 - 9.44iT - 43T^{2} \)
47 \( 1 - 0.242iT - 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 2.22iT - 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 + 1.94T + 67T^{2} \)
71 \( 1 - 7.57iT - 71T^{2} \)
73 \( 1 + 3.49iT - 73T^{2} \)
79 \( 1 + 11.6iT - 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 - 2.58iT - 89T^{2} \)
97 \( 1 + 5.06T + 97T^{2} \)
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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

−12.33258002311079853618175414378, −10.72812959335873851620954100882, −9.644480968680631584892640622944, −8.964449551334843711663883112764, −8.136039981610187423227662260558, −7.60301186429814031204619254318, −5.61461253124997088012701433807, −4.56606102278166294002226250580, −2.92963405560250436927423347351, −0.990742756082754741049501171350, 2.24697676654091503666568136951, 3.75323016016836443040769332717, 4.78871401190560469938064964657, 6.74814900469536910950841254611, 7.67676896657413871564758246018, 8.542171034157081905609610912949, 9.576194907943516264388337124386, 10.30988786512695525528325498297, 10.93910707726691248115024974918, 12.58991684756089806186728981411

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