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Properties

Label 2-231-21.20-c1-0-11
Degree $2$
Conductor $231$
Sign $0.992 + 0.118i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.992 + 0.118i$
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: \((2,\ 231,\ (\ :1/2),\ 0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58343 - 0.0941307i\)
\(L(\frac12)\) \(\approx\) \(1.58343 - 0.0941307i\)
\(L(\frac{3}{2})\) 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

−11.78908232677240029469731613094, −11.33371478313242258427134775254, −10.30068440075118251982441006310, −9.555797564078608080227301618894, −8.355818989581698591235697099135, −7.47137395708560238384599105639, −5.75635633596484945372940391936, −4.46499358908162635377576900094, −3.32363900593501957153526798606, −2.06727642916746643405930607277, 1.69292019141852943145342145019, 3.40714185770863885480299021350, 5.17468911993657772533028451757, 6.40137000534073334087337064687, 7.34770560578140110507190336776, 7.959243284513055078529241951683, 8.744103985409679458888848868399, 10.39235570821025066248661003705, 11.54726282721675009583189687580, 12.05364300728460851665903224774

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