Properties

Label 2-231-231.65-c1-0-25
Degree $2$
Conductor $231$
Sign $-0.672 + 0.739i$
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  + (0.939 − 1.62i)2-s + (1.16 − 1.27i)3-s + (−0.765 − 1.32i)4-s + (−2.63 − 1.52i)5-s + (−0.986 − 3.10i)6-s + (−0.738 + 2.54i)7-s + 0.882·8-s + (−0.276 − 2.98i)9-s + (−4.95 + 2.86i)10-s + (0.988 − 3.16i)11-s + (−2.58 − 0.567i)12-s + 1.24i·13-s + (3.44 + 3.58i)14-s + (−5.02 + 1.59i)15-s + (2.35 − 4.08i)16-s + (2.83 + 4.90i)17-s + ⋯
L(s)  = 1  + (0.664 − 1.15i)2-s + (0.673 − 0.739i)3-s + (−0.382 − 0.662i)4-s + (−1.17 − 0.680i)5-s + (−0.402 − 1.26i)6-s + (−0.279 + 0.960i)7-s + 0.311·8-s + (−0.0922 − 0.995i)9-s + (−1.56 + 0.904i)10-s + (0.298 − 0.954i)11-s + (−0.747 − 0.163i)12-s + 0.345i·13-s + (0.919 + 0.959i)14-s + (−1.29 + 0.412i)15-s + (0.589 − 1.02i)16-s + (0.687 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.672 + 0.739i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.672 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.729958 - 1.65030i\)
\(L(\frac12)\) \(\approx\) \(0.729958 - 1.65030i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 + 1.27i)T \)
7 \( 1 + (0.738 - 2.54i)T \)
11 \( 1 + (-0.988 + 3.16i)T \)
good2 \( 1 + (-0.939 + 1.62i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.63 + 1.52i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.24iT - 13T^{2} \)
17 \( 1 + (-2.83 - 4.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.532 - 0.307i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.41 - 2.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.47T + 29T^{2} \)
31 \( 1 + (-3.18 - 5.50i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.15 - 5.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 2.31iT - 43T^{2} \)
47 \( 1 + (9.35 + 5.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.58 + 1.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.38 + 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.24 - 4.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 4.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + (1.01 - 0.586i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.34 + 1.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + (-1.19 - 0.690i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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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

−11.79056907879490697562891352570, −11.63949710552088489261473831529, −10.05167608318306182334696821466, −8.614816176539228564611998274143, −8.283778201672970038346541944262, −6.82299248423881468173147075297, −5.28660246912753813113130767135, −3.77795773017146709907366489085, −3.09495287249740806832153014345, −1.41364541407887580357033832293, 3.21374460408826366667903382291, 4.19153922362859770504575635284, 5.02528125883150800398956167095, 6.82506695386303191084864104286, 7.39336525960282824913658902170, 8.145968166154250933000490017872, 9.690512105892069144302839037392, 10.53366502922958756548771936959, 11.52914793614506944772923581649, 12.92169486129987775480105415553

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