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Properties

Label 2-231-21.20-c1-0-15
Degree $2$
Conductor $231$
Sign $0.970 - 0.242i$
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  + 0.309i·2-s + (1.65 − 0.502i)3-s + 1.90·4-s − 1.52·5-s + (0.155 + 0.513i)6-s + (−0.131 + 2.64i)7-s + 1.20i·8-s + (2.49 − 1.66i)9-s − 0.473i·10-s + i·11-s + (3.15 − 0.957i)12-s − 1.78i·13-s + (−0.818 − 0.0406i)14-s + (−2.53 + 0.768i)15-s + 3.43·16-s − 5.34·17-s + ⋯
L(s)  = 1  + 0.219i·2-s + (0.956 − 0.290i)3-s + 0.952·4-s − 0.684·5-s + (0.0635 + 0.209i)6-s + (−0.0496 + 0.998i)7-s + 0.427i·8-s + (0.831 − 0.555i)9-s − 0.149i·10-s + 0.301i·11-s + (0.911 − 0.276i)12-s − 0.495i·13-s + (−0.218 − 0.0108i)14-s + (−0.654 + 0.198i)15-s + 0.858·16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.970 - 0.242i$
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: \((2,\ 231,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75072 + 0.215396i\)
\(L(\frac12)\) \(\approx\) \(1.75072 + 0.215396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 0.502i)T \)
7 \( 1 + (0.131 - 2.64i)T \)
11 \( 1 - iT \)
good2 \( 1 - 0.309iT - 2T^{2} \)
5 \( 1 + 1.52T + 5T^{2} \)
13 \( 1 + 1.78iT - 13T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 + 5.87iT - 19T^{2} \)
23 \( 1 + 7.34iT - 23T^{2} \)
29 \( 1 - 6.99iT - 29T^{2} \)
31 \( 1 - 7.96iT - 31T^{2} \)
37 \( 1 + 1.02T + 37T^{2} \)
41 \( 1 - 1.69T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 7.18T + 47T^{2} \)
53 \( 1 + 9.97iT - 53T^{2} \)
59 \( 1 + 6.79T + 59T^{2} \)
61 \( 1 - 2.66iT - 61T^{2} \)
67 \( 1 + 2.91T + 67T^{2} \)
71 \( 1 - 8.60iT - 71T^{2} \)
73 \( 1 - 4.48iT - 73T^{2} \)
79 \( 1 - 3.23T + 79T^{2} \)
83 \( 1 - 8.97T + 83T^{2} \)
89 \( 1 - 0.618T + 89T^{2} \)
97 \( 1 + 9.26iT - 97T^{2} \)
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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

−12.32054873693930476616649677444, −11.39901716669500926773684808826, −10.37105909209986175281552243041, −8.895483213633054855452991927910, −8.383570314517217856275666612515, −7.16829760049328209538788611678, −6.52451466124334688354311669794, −4.85294819728156500573854855177, −3.16215405162945265193278222636, −2.18081227689579825982164835233, 1.91330771933772988580810855990, 3.50255418022016394974686211082, 4.21527190218117030058897391816, 6.24505703416499786838037429878, 7.51031637169360521168066794254, 7.901192224655249389724382643797, 9.372259052164487255059739891749, 10.28101992861992328127793657823, 11.18928189031008236213783241717, 11.95226262504268160611166305874

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