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Properties

Label 2-231-77.10-c1-0-10
Degree $2$
Conductor $231$
Sign $0.906 - 0.422i$
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 + (2.90 + 1.60i)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 + (0.875 + 0.482i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.906 - 0.422i$
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: \((2,\ 231,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30772 + 0.289795i\)
\(L(\frac12)\) \(\approx\) \(1.30772 + 0.289795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.90 - 1.60i)T \)
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

−12.73224026330973074911445965642, −11.12195588223713504794302838310, −9.912463287019013661286739790212, −9.318429131562952169017032636761, −8.531038852307969538597975574779, −7.46287949636196975504138886161, −6.28292919583642909727091934098, −4.66972777791919699939905680889, −3.79830953210380920694660593810, −1.72029652482038247291295068036, 1.74439067673243140715455220515, 2.82459483151111703371195831204, 4.99502014372591384444017869502, 6.04409829776266632856636301744, 6.93089849865676779162921465446, 8.805632848142299350431596277357, 9.048272010289087460816566968553, 10.05486454595514491517130839254, 10.99799035289203589559540030532, 11.99899036818474258001675228209

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