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Properties

Label 2-231-33.32-c1-0-12
Degree $2$
Conductor $231$
Sign $0.384 + 0.923i$
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  − 1.92·2-s + (1.23 − 1.21i)3-s + 1.69·4-s + 0.413i·5-s + (−2.37 + 2.33i)6-s − i·7-s + 0.588·8-s + (0.0504 − 2.99i)9-s − 0.794i·10-s + (3.07 + 1.23i)11-s + (2.09 − 2.05i)12-s − 1.35i·13-s + 1.92i·14-s + (0.502 + 0.510i)15-s − 4.51·16-s − 1.64·17-s + ⋯
L(s)  = 1  − 1.35·2-s + (0.713 − 0.701i)3-s + 0.846·4-s + 0.184i·5-s + (−0.969 + 0.952i)6-s − 0.377i·7-s + 0.208·8-s + (0.0168 − 0.999i)9-s − 0.251i·10-s + (0.927 + 0.372i)11-s + (0.603 − 0.593i)12-s − 0.376i·13-s + 0.513i·14-s + (0.129 + 0.131i)15-s − 1.12·16-s − 0.397·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.384 + 0.923i$
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: \((2,\ 231,\ (\ :1/2),\ 0.384 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658367 - 0.438904i\)
\(L(\frac12)\) \(\approx\) \(0.658367 - 0.438904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.23 + 1.21i)T \)
7 \( 1 + iT \)
11 \( 1 + (-3.07 - 1.23i)T \)
good2 \( 1 + 1.92T + 2T^{2} \)
5 \( 1 - 0.413iT - 5T^{2} \)
13 \( 1 + 1.35iT - 13T^{2} \)
17 \( 1 + 1.64T + 17T^{2} \)
19 \( 1 + 4.21iT - 19T^{2} \)
23 \( 1 + 6.16iT - 23T^{2} \)
29 \( 1 - 2.73T + 29T^{2} \)
31 \( 1 - 4.56T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 + 3.57T + 41T^{2} \)
43 \( 1 - 1.20iT - 43T^{2} \)
47 \( 1 - 8.33iT - 47T^{2} \)
53 \( 1 + 5.91iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 - 6.12T + 67T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 - 0.290iT - 73T^{2} \)
79 \( 1 - 16.6iT - 79T^{2} \)
83 \( 1 - 3.53T + 83T^{2} \)
89 \( 1 + 9.17iT - 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−11.89545499449346808830760948341, −10.77416129189528617600637175245, −9.867311124852108269227607823207, −8.878769765859840080840845460657, −8.341525168683513311000716507014, −7.09873642144494582792197804723, −6.65782621713339001363837047010, −4.39812681838996595353314725985, −2.63407375737939436832780868065, −1.05724140751640546116089913647, 1.77796474857755981839812270710, 3.57054564863945961670456849020, 4.94997403154528390363524812191, 6.63793771257194870443982996842, 7.915752827870842726357075556723, 8.705381199034276357343914291220, 9.300015858890640975055612601072, 10.11776561958577792501098923971, 11.05145670890819815156071504071, 12.05262744018034847112385315177

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