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Properties

Label 2-231-11.4-c1-0-1
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: $\chi_{231} (169, \cdot )$
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

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

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