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Properties

Label 2-231-231.65-c1-0-0
Degree $2$
Conductor $231$
Sign $-0.801 - 0.598i$
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  + (−0.468 + 0.811i)2-s + (−0.688 − 1.58i)3-s + (0.560 + 0.971i)4-s + (−2.73 − 1.58i)5-s + (1.61 + 0.185i)6-s + (0.0402 + 2.64i)7-s − 2.92·8-s + (−2.05 + 2.18i)9-s + (2.56 − 1.48i)10-s + (−0.269 + 3.30i)11-s + (1.15 − 1.56i)12-s + 1.78i·13-s + (−2.16 − 1.20i)14-s + (−0.626 + 5.44i)15-s + (0.249 − 0.431i)16-s + (2.35 + 4.07i)17-s + ⋯
L(s)  = 1  + (−0.331 + 0.573i)2-s + (−0.397 − 0.917i)3-s + (0.280 + 0.485i)4-s + (−1.22 − 0.707i)5-s + (0.658 + 0.0757i)6-s + (0.0151 + 0.999i)7-s − 1.03·8-s + (−0.683 + 0.729i)9-s + (0.811 − 0.468i)10-s + (−0.0813 + 0.996i)11-s + (0.334 − 0.450i)12-s + 0.494i·13-s + (−0.578 − 0.322i)14-s + (−0.161 + 1.40i)15-s + (0.0622 − 0.107i)16-s + (0.570 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.801 - 0.598i$
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: \((2,\ 231,\ (\ :1/2),\ -0.801 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127437 + 0.383868i\)
\(L(\frac12)\) \(\approx\) \(0.127437 + 0.383868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.688 + 1.58i)T \)
7 \( 1 + (-0.0402 - 2.64i)T \)
11 \( 1 + (0.269 - 3.30i)T \)
good2 \( 1 + (0.468 - 0.811i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.73 + 1.58i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.78iT - 13T^{2} \)
17 \( 1 + (-2.35 - 4.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.24 + 3.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.462 - 0.267i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.46T + 29T^{2} \)
31 \( 1 + (1.71 + 2.96i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.08 + 3.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.272T + 41T^{2} \)
43 \( 1 - 5.32iT - 43T^{2} \)
47 \( 1 + (-7.52 - 4.34i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.61 + 5.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.88 - 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.14 + 0.659i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.766 - 1.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.2iT - 71T^{2} \)
73 \( 1 + (-0.516 + 0.298i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.5 - 6.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 + (7.21 + 4.16i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.02T + 97T^{2} \)
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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

−12.62759296726581762486150216014, −11.82662171926429422186441560093, −11.10461099479828895473421987671, −9.151512114925390609520950157781, −8.361357203390585231480308456379, −7.67480719007082275944874937237, −6.76840444226433397240887982228, −5.64646311762776250508857786698, −4.16348580579569993924850164844, −2.26585923082746378616193844606, 0.36751375055053531078824786578, 3.14623492844534932014916115831, 3.96426607397417525308914073770, 5.53039012934033538658636604876, 6.71469900203335439721579280729, 7.919758118165588492857895022096, 9.150655057479393601956251887692, 10.48308535474656439341992693301, 10.63479156850804329758914241790, 11.45033930401584499428423642843

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