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Properties

Label 2-231-231.230-c2-0-49
Degree $2$
Conductor $231$
Sign $-0.610 + 0.791i$
Analytic cond. $6.29429$
Root an. cond. $2.50884$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.01·2-s + (−2.47 + 1.70i)3-s + 0.0592·4-s + 0.175·5-s + (−4.97 + 3.43i)6-s + (−5.46 − 4.37i)7-s − 7.93·8-s + (3.20 − 8.41i)9-s + 0.354·10-s + (6.07 − 9.17i)11-s + (−0.146 + 0.100i)12-s − 7.53·13-s + (−11.0 − 8.82i)14-s + (−0.434 + 0.299i)15-s − 16.2·16-s − 21.7i·17-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.823 + 0.567i)3-s + 0.0148·4-s + 0.0351·5-s + (−0.829 + 0.571i)6-s + (−0.780 − 0.625i)7-s − 0.992·8-s + (0.355 − 0.934i)9-s + 0.0354·10-s + (0.552 − 0.833i)11-s + (−0.0122 + 0.00841i)12-s − 0.579·13-s + (−0.786 − 0.630i)14-s + (−0.0289 + 0.0199i)15-s − 1.01·16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.610 + 0.791i$
Analytic conductor: \(6.29429\)
Root analytic conductor: \(2.50884\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1),\ -0.610 + 0.791i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.264315 - 0.537674i\)
\(L(\frac12)\) \(\approx\) \(0.264315 - 0.537674i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.47 - 1.70i)T \)
7 \( 1 + (5.46 + 4.37i)T \)
11 \( 1 + (-6.07 + 9.17i)T \)
good2 \( 1 - 2.01T + 4T^{2} \)
5 \( 1 - 0.175T + 25T^{2} \)
13 \( 1 + 7.53T + 169T^{2} \)
17 \( 1 + 21.7iT - 289T^{2} \)
19 \( 1 + 13.1T + 361T^{2} \)
23 \( 1 - 28.3iT - 529T^{2} \)
29 \( 1 - 7.12T + 841T^{2} \)
31 \( 1 - 0.122iT - 961T^{2} \)
37 \( 1 + 53.3T + 1.36e3T^{2} \)
41 \( 1 + 55.3iT - 1.68e3T^{2} \)
43 \( 1 - 13.0iT - 1.84e3T^{2} \)
47 \( 1 - 48.1T + 2.20e3T^{2} \)
53 \( 1 - 35.9iT - 2.80e3T^{2} \)
59 \( 1 + 81.5T + 3.48e3T^{2} \)
61 \( 1 + 34.1T + 3.72e3T^{2} \)
67 \( 1 - 84.8T + 4.48e3T^{2} \)
71 \( 1 + 11.1iT - 5.04e3T^{2} \)
73 \( 1 - 106.T + 5.32e3T^{2} \)
79 \( 1 + 37.9iT - 6.24e3T^{2} \)
83 \( 1 + 56.2iT - 6.88e3T^{2} \)
89 \( 1 + 14.4T + 7.92e3T^{2} \)
97 \( 1 + 133. iT - 9.40e3T^{2} \)
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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

−11.85739886891217922363274936328, −10.86207782400643693768832518689, −9.719858856290207066369266580691, −9.059262485994935217969463846048, −7.16547012411411526472753284230, −6.15313718701837675313287632195, −5.28508074566859074213965117323, −4.14616396357225460956183681918, −3.29052228632616258909207403926, −0.24795922020966014915611728210, 2.24513546373301021648986727317, 3.98212666723346925054558584277, 5.06086938250566786411075437741, 6.16153010524888745800192801855, 6.71419419000072163584256106333, 8.296603360422540201244724896412, 9.520697021873544061984135079799, 10.54540086143685216031581087657, 11.92909565567358464637238982823, 12.44747570667199368878082301710

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