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Properties

Label 2-231-21.20-c1-0-5
Degree $2$
Conductor $231$
Sign $-0.620 + 0.783i$
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  + 2.73i·2-s + (−0.280 + 1.70i)3-s − 5.49·4-s + 2.40·5-s + (−4.68 − 0.768i)6-s + (−1.28 + 2.31i)7-s − 9.58i·8-s + (−2.84 − 0.959i)9-s + 6.58i·10-s + i·11-s + (1.54 − 9.39i)12-s − 1.84i·13-s + (−6.33 − 3.51i)14-s + (−0.675 + 4.11i)15-s + 15.2·16-s + 5.51·17-s + ⋯
L(s)  = 1  + 1.93i·2-s + (−0.162 + 0.986i)3-s − 2.74·4-s + 1.07·5-s + (−1.91 − 0.313i)6-s + (−0.485 + 0.874i)7-s − 3.38i·8-s + (−0.947 − 0.319i)9-s + 2.08i·10-s + 0.301i·11-s + (0.445 − 2.71i)12-s − 0.511i·13-s + (−1.69 − 0.940i)14-s + (−0.174 + 1.06i)15-s + 3.80·16-s + 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.620 + 0.783i$
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: \((2,\ 231,\ (\ :1/2),\ -0.620 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439504 - 0.908673i\)
\(L(\frac12)\) \(\approx\) \(0.439504 - 0.908673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.280 - 1.70i)T \)
7 \( 1 + (1.28 - 2.31i)T \)
11 \( 1 - iT \)
good2 \( 1 - 2.73iT - 2T^{2} \)
5 \( 1 - 2.40T + 5T^{2} \)
13 \( 1 + 1.84iT - 13T^{2} \)
17 \( 1 - 5.51T + 17T^{2} \)
19 \( 1 - 4.50iT - 19T^{2} \)
23 \( 1 - 0.776iT - 23T^{2} \)
29 \( 1 - 2.13iT - 29T^{2} \)
31 \( 1 - 1.00iT - 31T^{2} \)
37 \( 1 - 0.563T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 + 4.25T + 47T^{2} \)
53 \( 1 + 6.96iT - 53T^{2} \)
59 \( 1 - 1.34T + 59T^{2} \)
61 \( 1 - 11.1iT - 61T^{2} \)
67 \( 1 - 6.52T + 67T^{2} \)
71 \( 1 + 5.51iT - 71T^{2} \)
73 \( 1 - 6.97iT - 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 - 1.17T + 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−13.15127614264540906059738564930, −12.19689191624128754849450052507, −10.11384253655539435282191778804, −9.754082313920851875313541711035, −8.859902603039424350107623521078, −7.896900651493057234237693838809, −6.39538558407897052275623347158, −5.67138594755806061303383300965, −5.13275839529383926512148793395, −3.52240973713617835768422481291, 0.920119266850076553646639729094, 2.21139176651038476495148191314, 3.42040434830967975921253441238, 5.02549519512130091689647004237, 6.27409160354526378186170975286, 7.82976125697094806746929674863, 9.129492569548747442275878591615, 9.867957794457513150650132909356, 10.75272735025083803753416397718, 11.59733547094705543247990088647

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