Properties

Label 2-231-1.1-c1-0-5
Degree $2$
Conductor $231$
Sign $1$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93·2-s + 3-s + 1.74·4-s + 4.18·5-s − 1.93·6-s − 7-s + 0.491·8-s + 9-s − 8.10·10-s − 11-s + 1.74·12-s − 3.17·13-s + 1.93·14-s + 4.18·15-s − 4.44·16-s + 6.85·17-s − 1.93·18-s − 0.318·19-s + 7.31·20-s − 21-s + 1.93·22-s − 1.87·23-s + 0.491·24-s + 12.5·25-s + 6.14·26-s + 27-s − 1.74·28-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.577·3-s + 0.872·4-s + 1.87·5-s − 0.790·6-s − 0.377·7-s + 0.173·8-s + 0.333·9-s − 2.56·10-s − 0.301·11-s + 0.503·12-s − 0.880·13-s + 0.517·14-s + 1.08·15-s − 1.11·16-s + 1.66·17-s − 0.456·18-s − 0.0731·19-s + 1.63·20-s − 0.218·21-s + 0.412·22-s − 0.390·23-s + 0.100·24-s + 2.51·25-s + 1.20·26-s + 0.192·27-s − 0.329·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.9621394117\)
\(L(\frac12)\) \(\approx\) \(0.9621394117\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + 1.93T + 2T^{2} \)
5 \( 1 - 4.18T + 5T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 + 0.318T + 19T^{2} \)
23 \( 1 + 1.87T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 - 9.23T + 31T^{2} \)
37 \( 1 + 7.55T + 37T^{2} \)
41 \( 1 - 9.36T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 - 0.508T + 53T^{2} \)
59 \( 1 + 7.04T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 2.66T + 67T^{2} \)
71 \( 1 + 5.01T + 71T^{2} \)
73 \( 1 + 4.82T + 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + 1.74T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−12.21456910915472715821731424418, −10.52997061934434268445770386664, −9.862077681512781941673610626796, −9.586722059945123837281668420209, −8.497173234860566319854544466412, −7.47539022279127247059199233696, −6.33252302571214420850142548681, −5.08278705547917041221747902990, −2.79494867967071850087724802645, −1.57297212451302457402103914109, 1.57297212451302457402103914109, 2.79494867967071850087724802645, 5.08278705547917041221747902990, 6.33252302571214420850142548681, 7.47539022279127247059199233696, 8.497173234860566319854544466412, 9.586722059945123837281668420209, 9.862077681512781941673610626796, 10.52997061934434268445770386664, 12.21456910915472715821731424418

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