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Properties

Label 2-231-77.54-c1-0-2
Degree $2$
Conductor $231$
Sign $0.731 + 0.681i$
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  + (−1.49 − 0.863i)2-s + (−0.866 + 0.5i)3-s + (0.489 + 0.848i)4-s + (1.43 + 0.827i)5-s + 1.72·6-s + (−2.00 − 1.72i)7-s + 1.76i·8-s + (0.499 − 0.866i)9-s + (−1.42 − 2.47i)10-s + (3.00 + 1.40i)11-s + (−0.848 − 0.489i)12-s + 2.97·13-s + (1.51 + 4.30i)14-s − 1.65·15-s + (2.49 − 4.32i)16-s + (0.747 + 1.29i)17-s + ⋯
L(s)  = 1  + (−1.05 − 0.610i)2-s + (−0.499 + 0.288i)3-s + (0.244 + 0.424i)4-s + (0.641 + 0.370i)5-s + 0.704·6-s + (−0.758 − 0.651i)7-s + 0.622i·8-s + (0.166 − 0.288i)9-s + (−0.451 − 0.782i)10-s + (0.905 + 0.424i)11-s + (−0.244 − 0.141i)12-s + 0.825·13-s + (0.404 + 1.15i)14-s − 0.427·15-s + (0.624 − 1.08i)16-s + (0.181 + 0.314i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.731 + 0.681i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617931 - 0.243299i\)
\(L(\frac12)\) \(\approx\) \(0.617931 - 0.243299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.00 + 1.72i)T \)
11 \( 1 + (-3.00 - 1.40i)T \)
good2 \( 1 + (1.49 + 0.863i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.43 - 0.827i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 2.97T + 13T^{2} \)
17 \( 1 + (-0.747 - 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.75 + 4.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.31 + 7.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.02iT - 29T^{2} \)
31 \( 1 + (-7.67 + 4.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.11 - 7.12i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.12T + 41T^{2} \)
43 \( 1 - 5.03iT - 43T^{2} \)
47 \( 1 + (6.38 + 3.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.630 - 1.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.30 - 2.48i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.82 + 3.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 + (4.92 + 8.53i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.41 + 1.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (4.47 + 2.58i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.35iT - 97T^{2} \)
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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

−11.68427877133944505293928621249, −10.82458664695033746546587799483, −10.15593118987144401080594243284, −9.463228323517737597019174555071, −8.571236059366523425719677365126, −6.96247695980183035955007196726, −6.17932039370592447326438077971, −4.61393821271496541130169915886, −2.95285854823499693053590681636, −1.08158108738211367372561964321, 1.25579079646554035313067589911, 3.58605487099115882565966123214, 5.65155624278068900800471159306, 6.23593439010056721500981287619, 7.34449355836542927065215977956, 8.505299088223361281452199235737, 9.375513058855678428766046589484, 9.898661783042750199731484841507, 11.32673080362839277015238541761, 12.27552829448667193859796483890

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