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Properties

Label 2-231-21.20-c1-0-20
Degree $2$
Conductor $231$
Sign $-0.472 + 0.881i$
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  − 1.12i·2-s + (−0.799 − 1.53i)3-s + 0.743·4-s + 0.911·5-s + (−1.72 + 0.896i)6-s + (2.18 − 1.49i)7-s − 3.07i·8-s + (−1.72 + 2.45i)9-s − 1.02i·10-s + i·11-s + (−0.594 − 1.14i)12-s + 0.687i·13-s + (−1.67 − 2.44i)14-s + (−0.728 − 1.40i)15-s − 1.95·16-s − 3.26·17-s + ⋯
L(s)  = 1  − 0.792i·2-s + (−0.461 − 0.887i)3-s + 0.371·4-s + 0.407·5-s + (−0.703 + 0.365i)6-s + (0.826 − 0.563i)7-s − 1.08i·8-s + (−0.573 + 0.819i)9-s − 0.322i·10-s + 0.301i·11-s + (−0.171 − 0.329i)12-s + 0.190i·13-s + (−0.446 − 0.654i)14-s + (−0.188 − 0.361i)15-s − 0.489·16-s − 0.792·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.472 + 0.881i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.686573 - 1.14737i\)
\(L(\frac12)\) \(\approx\) \(0.686573 - 1.14737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.799 + 1.53i)T \)
7 \( 1 + (-2.18 + 1.49i)T \)
11 \( 1 - iT \)
good2 \( 1 + 1.12iT - 2T^{2} \)
5 \( 1 - 0.911T + 5T^{2} \)
13 \( 1 - 0.687iT - 13T^{2} \)
17 \( 1 + 3.26T + 17T^{2} \)
19 \( 1 - 0.717iT - 19T^{2} \)
23 \( 1 + 0.687iT - 23T^{2} \)
29 \( 1 - 0.255iT - 29T^{2} \)
31 \( 1 - 8.80iT - 31T^{2} \)
37 \( 1 + 3.81T + 37T^{2} \)
41 \( 1 - 9.59T + 41T^{2} \)
43 \( 1 - 8.59T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 - 5.09iT - 53T^{2} \)
59 \( 1 + 5.86T + 59T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 + 8.19T + 67T^{2} \)
71 \( 1 - 5.71iT - 71T^{2} \)
73 \( 1 + 9.05iT - 73T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 - 3.94T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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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

−11.92132224879012340222882893458, −10.91007252665876795519519560103, −10.50138988176398740032879494463, −9.049112988952389488234768122043, −7.65474504807319799151530255111, −6.91483430601659712353470649747, −5.78633796309106358971144672323, −4.31130673846230964884865484681, −2.44458079583611511081223737544, −1.35582451602331290539730209509, 2.41862162179696500524603622977, 4.34915573433443331209287616652, 5.58718913176145903672271543986, 6.06032494759853996513599279261, 7.49214370207104357549423330130, 8.588010687605035318456703424239, 9.481293354334362833994938366928, 10.83303893560911115333488489271, 11.30316365971641013901599610516, 12.26932324987033487276020704660

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