Properties

Label 2-231-11.3-c1-0-5
Degree $2$
Conductor $231$
Sign $0.894 - 0.447i$
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  + (−2.13 + 1.55i)2-s + (0.309 + 0.951i)3-s + (1.53 − 4.72i)4-s + (−2.49 − 1.81i)5-s + (−2.13 − 1.55i)6-s + (−0.309 + 0.951i)7-s + (2.42 + 7.45i)8-s + (−0.809 + 0.587i)9-s + 8.14·10-s + (1.45 − 2.98i)11-s + 4.96·12-s + (4.56 − 3.31i)13-s + (−0.815 − 2.51i)14-s + (0.952 − 2.93i)15-s + (−8.70 − 6.32i)16-s + (4.53 + 3.29i)17-s + ⋯
L(s)  = 1  + (−1.51 + 1.09i)2-s + (0.178 + 0.549i)3-s + (0.767 − 2.36i)4-s + (−1.11 − 0.810i)5-s + (−0.871 − 0.633i)6-s + (−0.116 + 0.359i)7-s + (0.856 + 2.63i)8-s + (−0.269 + 0.195i)9-s + 2.57·10-s + (0.438 − 0.898i)11-s + 1.43·12-s + (1.26 − 0.919i)13-s + (−0.218 − 0.671i)14-s + (0.246 − 0.757i)15-s + (−2.17 − 1.58i)16-s + (1.09 + 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.512967 + 0.121304i\)
\(L(\frac12)\) \(\approx\) \(0.512967 + 0.121304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-1.45 + 2.98i)T \)
good2 \( 1 + (2.13 - 1.55i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (2.49 + 1.81i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.56 + 3.31i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.53 - 3.29i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0379 - 0.116i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.67T + 23T^{2} \)
29 \( 1 + (-2.73 + 8.41i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.29 - 0.939i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.28 + 3.96i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.07 - 3.31i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.10T + 43T^{2} \)
47 \( 1 + (0.492 + 1.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.59 + 6.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.04 + 6.29i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.86 - 4.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.04T + 67T^{2} \)
71 \( 1 + (5.05 + 3.67i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.08 + 3.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.60 + 5.52i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.63 - 5.54i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.45T + 89T^{2} \)
97 \( 1 + (-4.59 + 3.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.89159391159871567737139829082, −10.97059833308189740513731836950, −10.05324692858186243747291145363, −8.941120641119414616913293502805, −8.312391619215349221807462984115, −7.87821001317180372383760782212, −6.25733512997718449042122382963, −5.41054223561132007014542998328, −3.70488770365727301035004920386, −0.842098642116589492520257484617, 1.31832354572488906208429765563, 3.00304454966175892501691603821, 3.93056104599807862193991764083, 6.89552068778776400722821010413, 7.28598370046274693102612940037, 8.331591309487685499706779265986, 9.225564681744568341300175140995, 10.29166033073565609930860736452, 11.18171607577897985887346456924, 11.79589673510884719457172543025

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