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Properties

Label 2-231-77.54-c1-0-10
Degree $2$
Conductor $231$
Sign $0.922 + 0.386i$
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  + (0.533 + 0.307i)2-s + (0.866 − 0.5i)3-s + (−0.810 − 1.40i)4-s + (2.84 + 1.64i)5-s + 0.615·6-s + (−0.613 − 2.57i)7-s − 2.22i·8-s + (0.499 − 0.866i)9-s + (1.01 + 1.75i)10-s + (−2.83 + 1.71i)11-s + (−1.40 − 0.810i)12-s + 0.142·13-s + (0.465 − 1.56i)14-s + 3.28·15-s + (−0.935 + 1.61i)16-s + (3.63 + 6.29i)17-s + ⋯
L(s)  = 1  + (0.376 + 0.217i)2-s + (0.499 − 0.288i)3-s + (−0.405 − 0.701i)4-s + (1.27 + 0.734i)5-s + 0.251·6-s + (−0.231 − 0.972i)7-s − 0.788i·8-s + (0.166 − 0.288i)9-s + (0.319 + 0.553i)10-s + (−0.855 + 0.517i)11-s + (−0.405 − 0.233i)12-s + 0.0395·13-s + (0.124 − 0.417i)14-s + 0.848·15-s + (−0.233 + 0.404i)16-s + (0.880 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.922 + 0.386i$
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: \((2,\ 231,\ (\ :1/2),\ 0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74116 - 0.349833i\)
\(L(\frac12)\) \(\approx\) \(1.74116 - 0.349833i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.613 + 2.57i)T \)
11 \( 1 + (2.83 - 1.71i)T \)
good2 \( 1 + (-0.533 - 0.307i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-2.84 - 1.64i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.142T + 13T^{2} \)
17 \( 1 + (-3.63 - 6.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.434 + 0.752i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.78 + 3.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.70iT - 29T^{2} \)
31 \( 1 + (4.17 - 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.69 - 8.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.19T + 41T^{2} \)
43 \( 1 - 9.66iT - 43T^{2} \)
47 \( 1 + (7.58 + 4.38i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.31 - 5.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.33 - 0.773i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.62 + 13.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.50 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.66T + 71T^{2} \)
73 \( 1 + (2.74 + 4.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.10 - 3.52i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 + (-6.82 - 3.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.07iT - 97T^{2} \)
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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

−12.68405725948757639701294373598, −10.68883211715920334794883574373, −10.19898129278148334810310230715, −9.549239685112709641581131621668, −8.075295337218197307461497021216, −6.81453579347193612041669684937, −6.12422080023600761726224392554, −4.88781340143412711565174024247, −3.37096956250737960881315023385, −1.69971599304526610232772962021, 2.31488025942677038484393193875, 3.36800481339509387342102198837, 5.23624988367684712970524480445, 5.44228837317388590237058562806, 7.45365850067362320855717181083, 8.700194898777954277212389492105, 9.162365619257944641270339545475, 10.07999404640676837611221244756, 11.54138395970670413043112671246, 12.51661906990954919496863682602

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