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Properties

Label 2-231-231.65-c1-0-23
Degree $2$
Conductor $231$
Sign $-0.306 + 0.951i$
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  + (1.36 − 2.36i)2-s + (1.68 + 0.379i)3-s + (−2.72 − 4.72i)4-s + (1.96 + 1.13i)5-s + (3.20 − 3.47i)6-s + (−1.90 + 1.83i)7-s − 9.44·8-s + (2.71 + 1.28i)9-s + (5.37 − 3.10i)10-s + (−3.29 + 0.415i)11-s + (−2.81 − 9.02i)12-s − 0.322i·13-s + (1.74 + 7.01i)14-s + (2.89 + 2.66i)15-s + (−7.43 + 12.8i)16-s + (1.73 + 2.99i)17-s + ⋯
L(s)  = 1  + (0.965 − 1.67i)2-s + (0.975 + 0.219i)3-s + (−1.36 − 2.36i)4-s + (0.880 + 0.508i)5-s + (1.30 − 1.42i)6-s + (−0.719 + 0.694i)7-s − 3.33·8-s + (0.903 + 0.427i)9-s + (1.69 − 0.981i)10-s + (−0.992 + 0.125i)11-s + (−0.813 − 2.60i)12-s − 0.0894i·13-s + (0.465 + 1.87i)14-s + (0.747 + 0.688i)15-s + (−1.85 + 3.21i)16-s + (0.419 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.306 + 0.951i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40194 - 1.92526i\)
\(L(\frac12)\) \(\approx\) \(1.40194 - 1.92526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.379i)T \)
7 \( 1 + (1.90 - 1.83i)T \)
11 \( 1 + (3.29 - 0.415i)T \)
good2 \( 1 + (-1.36 + 2.36i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.96 - 1.13i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.322iT - 13T^{2} \)
17 \( 1 + (-1.73 - 2.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 + 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.67 + 1.54i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + (2.42 + 4.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.690 - 1.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.42T + 41T^{2} \)
43 \( 1 + 5.82iT - 43T^{2} \)
47 \( 1 + (-2.37 - 1.37i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.824 - 0.475i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.17 + 5.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0609 - 0.0351i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.262 - 0.455i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.71iT - 71T^{2} \)
73 \( 1 + (3.18 - 1.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.08 - 2.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.50T + 83T^{2} \)
89 \( 1 + (2.19 + 1.26i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−12.20708278736212687967876889776, −10.72016947213602300454861723896, −10.17464549647998919492985418948, −9.530475736183599245977609441116, −8.452670271126336207841181735921, −6.36053618465444385078141209547, −5.32117409458696556049816033909, −3.96572525287738950797719918489, −2.73575600281093508876096361470, −2.19885156344515975245023728350, 2.94961557375500328381244331334, 4.18676703078771068614772185354, 5.38108053226867333425915690260, 6.44189830311823477885513777775, 7.37527742718873631466478048627, 8.211490097067728900677297818754, 9.183768663580269234683579044286, 10.09894103895533880694895679671, 12.38771621295830652664179148546, 12.97570866378070425763305424934

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