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Properties

Label 2-231-77.54-c1-0-5
Degree $2$
Conductor $231$
Sign $0.387 - 0.921i$
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 + (−0.284 + 3.30i)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 + (−0.0856 + 0.996i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.387 - 0.921i$
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: \((2,\ 231,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61495 + 1.07281i\)
\(L(\frac12)\) \(\approx\) \(1.61495 + 1.07281i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.284 - 3.30i)T \)
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

−12.36311448782163596319090709220, −11.82449659397600713215447607788, −10.27550158306600807035595229858, −9.818934602218671728427975761959, −8.252661834614830475592924285250, −6.86957612619637746709692052672, −6.10546290272111683795767073759, −5.03071509937173894252642389896, −4.38840855187709740760911589027, −2.38262021390306586158971839125, 1.64282597469239828582650646349, 3.28600429787230580414118398346, 4.85171355417559498393948602109, 5.29011925870560405053309839302, 6.70385120714144660178570943186, 8.004972531704824071982422602080, 9.150910343757639506913751012843, 10.67535522363105251445817388774, 11.16606343141864785551047710121, 12.10364606218714408943632246670

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