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Properties

Label 2-231-77.10-c1-0-5
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: $\chi_{231} (10, \cdot )$
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

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

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