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Properties

Label 2-231-21.20-c1-0-3
Degree $2$
Conductor $231$
Sign $-0.700 - 0.714i$
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  + 1.68i·2-s + (−1.02 − 1.39i)3-s − 0.855·4-s − 1.45·5-s + (2.35 − 1.73i)6-s + (0.366 + 2.62i)7-s + 1.93i·8-s + (−0.881 + 2.86i)9-s − 2.46i·10-s + i·11-s + (0.880 + 1.19i)12-s + 3.51i·13-s + (−4.42 + 0.619i)14-s + (1.50 + 2.03i)15-s − 4.97·16-s + 2.75·17-s + ⋯
L(s)  = 1  + 1.19i·2-s + (−0.594 − 0.804i)3-s − 0.427·4-s − 0.652·5-s + (0.961 − 0.710i)6-s + (0.138 + 0.990i)7-s + 0.683i·8-s + (−0.293 + 0.955i)9-s − 0.779i·10-s + 0.301i·11-s + (0.254 + 0.344i)12-s + 0.975i·13-s + (−1.18 + 0.165i)14-s + (0.387 + 0.524i)15-s − 1.24·16-s + 0.667·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.700 - 0.714i$
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: \((2,\ 231,\ (\ :1/2),\ -0.700 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.331946 + 0.790227i\)
\(L(\frac12)\) \(\approx\) \(0.331946 + 0.790227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.02 + 1.39i)T \)
7 \( 1 + (-0.366 - 2.62i)T \)
11 \( 1 - iT \)
good2 \( 1 - 1.68iT - 2T^{2} \)
5 \( 1 + 1.45T + 5T^{2} \)
13 \( 1 - 3.51iT - 13T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 - 4.07iT - 19T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + 1.86iT - 29T^{2} \)
31 \( 1 + 5.81iT - 31T^{2} \)
37 \( 1 - 7.59T + 37T^{2} \)
41 \( 1 - 0.863T + 41T^{2} \)
43 \( 1 - 2.11T + 43T^{2} \)
47 \( 1 - 0.483T + 47T^{2} \)
53 \( 1 - 13.8iT - 53T^{2} \)
59 \( 1 + 5.87T + 59T^{2} \)
61 \( 1 - 5.37iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 8.75T + 79T^{2} \)
83 \( 1 - 7.13T + 83T^{2} \)
89 \( 1 - 15.2T + 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−12.28030990549715048134820611914, −11.88637787208278174084419319613, −10.93629146108403157073441657522, −9.293681076614500446807158984302, −8.055876184662623308462609689936, −7.63530266773777433344853458524, −6.38921287651888351701166383469, −5.79432194875517472041682705031, −4.53969384765850083224770769132, −2.21305093397576547567865311663, 0.77270924487916296217126715457, 3.21623256525530726824122967185, 3.96689481534076318257945080986, 5.20483571374980233985557184052, 6.75595747706655052123211172031, 7.954507469389832357133863401196, 9.453260221422644138474003749594, 10.22014919568723341376929799874, 11.02475432692142259534946737616, 11.48307043484562768036882131660

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