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Properties

Label 2-231-33.32-c1-0-8
Degree $2$
Conductor $231$
Sign $0.986 + 0.163i$
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  − 0.628·2-s + (−1.53 + 0.792i)3-s − 1.60·4-s − 1.39i·5-s + (0.968 − 0.498i)6-s + i·7-s + 2.26·8-s + (1.74 − 2.44i)9-s + 0.880i·10-s + (1.97 − 2.66i)11-s + (2.47 − 1.27i)12-s + 5.12i·13-s − 0.628i·14-s + (1.10 + 2.15i)15-s + 1.78·16-s + 7.86·17-s + ⋯
L(s)  = 1  − 0.444·2-s + (−0.889 + 0.457i)3-s − 0.802·4-s − 0.625i·5-s + (0.395 − 0.203i)6-s + 0.377i·7-s + 0.801·8-s + (0.580 − 0.814i)9-s + 0.278i·10-s + (0.596 − 0.802i)11-s + (0.713 − 0.367i)12-s + 1.42i·13-s − 0.168i·14-s + (0.286 + 0.556i)15-s + 0.445·16-s + 1.90·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652075 - 0.0536517i\)
\(L(\frac12)\) \(\approx\) \(0.652075 - 0.0536517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.792i)T \)
7 \( 1 - iT \)
11 \( 1 + (-1.97 + 2.66i)T \)
good2 \( 1 + 0.628T + 2T^{2} \)
5 \( 1 + 1.39iT - 5T^{2} \)
13 \( 1 - 5.12iT - 13T^{2} \)
17 \( 1 - 7.86T + 17T^{2} \)
19 \( 1 + 5.38iT - 19T^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + 2.12T + 29T^{2} \)
31 \( 1 + 2.48T + 31T^{2} \)
37 \( 1 - 1.31T + 37T^{2} \)
41 \( 1 - 6.47T + 41T^{2} \)
43 \( 1 + 4.29iT - 43T^{2} \)
47 \( 1 + 6.76iT - 47T^{2} \)
53 \( 1 + 7.79iT - 53T^{2} \)
59 \( 1 - 14.8iT - 59T^{2} \)
61 \( 1 - 9.28iT - 61T^{2} \)
67 \( 1 - 9.60T + 67T^{2} \)
71 \( 1 + 9.34iT - 71T^{2} \)
73 \( 1 - 9.47iT - 73T^{2} \)
79 \( 1 - 2.26iT - 79T^{2} \)
83 \( 1 + 5.64T + 83T^{2} \)
89 \( 1 + 3.80iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−12.03838440198373251648097670853, −11.24307487479547275297599441039, −10.08700814659543074970181823565, −9.179999453287023857092879654160, −8.696815343853457164131582278429, −7.14229185566410160903039438115, −5.78596454753901914491133933981, −4.87442715615861068098504148823, −3.82833702332122546987247738102, −0.967454612253655487439346676452, 1.19072546786836419672198887340, 3.60823333374626705339035460067, 5.06820178727816235836548731063, 6.05272157058113479629429455552, 7.52674135725388180323490560193, 7.87724374231471792465666513623, 9.673373059821548849509773632088, 10.21741126203409869355069486176, 11.07262311920819382705762538515, 12.45960571533440647811016028440

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