Properties

Label 2-231-77.10-c1-0-7
Degree $2$
Conductor $231$
Sign $0.631 - 0.775i$
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  + (−2.16 + 1.25i)2-s + (0.866 + 0.5i)3-s + (2.13 − 3.69i)4-s + (−0.849 + 0.490i)5-s − 2.50·6-s + (1.62 − 2.08i)7-s + 5.66i·8-s + (0.499 + 0.866i)9-s + (1.22 − 2.12i)10-s + (3.08 − 1.21i)11-s + (3.69 − 2.13i)12-s + 0.941·13-s + (−0.916 + 6.55i)14-s − 0.980·15-s + (−2.83 − 4.90i)16-s + (2.58 − 4.48i)17-s + ⋯
L(s)  = 1  + (−1.53 + 0.884i)2-s + (0.499 + 0.288i)3-s + (1.06 − 1.84i)4-s + (−0.379 + 0.219i)5-s − 1.02·6-s + (0.615 − 0.788i)7-s + 2.00i·8-s + (0.166 + 0.288i)9-s + (0.388 − 0.672i)10-s + (0.930 − 0.366i)11-s + (1.06 − 0.615i)12-s + 0.261·13-s + (−0.244 + 1.75i)14-s − 0.253·15-s + (−0.707 − 1.22i)16-s + (0.627 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.631 - 0.775i$
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: \((2,\ 231,\ (\ :1/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678263 + 0.322333i\)
\(L(\frac12)\) \(\approx\) \(0.678263 + 0.322333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 2.08i)T \)
11 \( 1 + (-3.08 + 1.21i)T \)
good2 \( 1 + (2.16 - 1.25i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (0.849 - 0.490i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.941T + 13T^{2} \)
17 \( 1 + (-2.58 + 4.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.27 - 2.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.379 - 0.656i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.74iT - 29T^{2} \)
31 \( 1 + (-5.50 - 3.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.49 + 2.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.80T + 41T^{2} \)
43 \( 1 + 6.73iT - 43T^{2} \)
47 \( 1 + (-8.42 + 4.86i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.24 + 3.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.7 + 7.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.81 - 4.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.21 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (-5.14 + 8.91i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.63 - 3.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.28T + 83T^{2} \)
89 \( 1 + (-2.06 + 1.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.30iT - 97T^{2} \)
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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

−11.91008464278760471872928650548, −10.94248534093478308498778466920, −10.15917643596550595751990445391, −9.188860467359892812215151806164, −8.422071236507154618137578321130, −7.46797500461382710926124854642, −6.86550081851941410940986975309, −5.32466492379001413929164376514, −3.61648527543470116560642247944, −1.28697538985140569924113354677, 1.36735721546250943836288982163, 2.62807559929672481715644100655, 4.15315896826328939043067448553, 6.28217589049136189828234945751, 7.77205081186264810072130097314, 8.256511557998957946211458790660, 9.141445254501133491649151367321, 9.910676574302696934765159551513, 11.09329946410199730597242793207, 11.98143547355700503636975783087

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