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Properties

Label 2-231-231.32-c1-0-17
Degree $2$
Conductor $231$
Sign $0.761 + 0.647i$
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  + (−0.463 − 0.803i)2-s + (1.07 + 1.35i)3-s + (0.569 − 0.986i)4-s + (1.53 − 0.886i)5-s + (0.590 − 1.49i)6-s + (2.26 − 1.36i)7-s − 2.91·8-s + (−0.680 + 2.92i)9-s + (−1.42 − 0.822i)10-s + (−3.22 + 0.791i)11-s + (1.95 − 0.289i)12-s − 0.151i·13-s + (−2.14 − 1.19i)14-s + (2.85 + 1.12i)15-s + (0.212 + 0.368i)16-s + (−0.598 + 1.03i)17-s + ⋯
L(s)  = 1  + (−0.328 − 0.568i)2-s + (0.621 + 0.783i)3-s + (0.284 − 0.493i)4-s + (0.686 − 0.396i)5-s + (0.241 − 0.610i)6-s + (0.857 − 0.514i)7-s − 1.02·8-s + (−0.226 + 0.973i)9-s + (−0.450 − 0.260i)10-s + (−0.971 + 0.238i)11-s + (0.563 − 0.0836i)12-s − 0.0421i·13-s + (−0.573 − 0.318i)14-s + (0.737 + 0.291i)15-s + (0.0531 + 0.0920i)16-s + (−0.145 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.761 + 0.647i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.761 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38823 - 0.510404i\)
\(L(\frac12)\) \(\approx\) \(1.38823 - 0.510404i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.07 - 1.35i)T \)
7 \( 1 + (-2.26 + 1.36i)T \)
11 \( 1 + (3.22 - 0.791i)T \)
good2 \( 1 + (0.463 + 0.803i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.53 + 0.886i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.151iT - 13T^{2} \)
17 \( 1 + (0.598 - 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.13 + 2.38i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.52 + 0.880i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.83T + 29T^{2} \)
31 \( 1 + (2.85 - 4.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 10.5iT - 43T^{2} \)
47 \( 1 + (-1.28 + 0.739i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.94 - 2.27i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.850 + 0.490i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.24 - 3.02i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.75 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.53iT - 71T^{2} \)
73 \( 1 + (9.61 + 5.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.02 + 4.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.653T + 83T^{2} \)
89 \( 1 + (-0.343 + 0.198i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.93T + 97T^{2} \)
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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

−11.79447679410188751116047791289, −10.72469896715252643572178924067, −10.30464132519296240414206791132, −9.350338514525173396493396762592, −8.512139788851842355890369287142, −7.27449354837838741902518230428, −5.52750491180248734966486824517, −4.77947313114691023970130980024, −2.99541541897128904032819822924, −1.68141957730140076029746400938, 2.11807063663803369736271006844, 3.17212336832817533793396821672, 5.40089237305695167309652254376, 6.44927025432118330677278820404, 7.48106757860382923288516948868, 8.195041474415099723841045287490, 8.988869966001429476454879117817, 10.22659514176497328491787023870, 11.63005635131676018838109973277, 12.23193988181191722066636114934

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