Properties

Label 2-231-77.10-c1-0-4
Degree $2$
Conductor $231$
Sign $-0.515 - 0.857i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.71 + 0.990i)2-s + (0.866 + 0.5i)3-s + (0.963 − 1.66i)4-s + (0.315 − 0.182i)5-s − 1.98·6-s + (0.143 + 2.64i)7-s − 0.145i·8-s + (0.499 + 0.866i)9-s + (−0.361 + 0.626i)10-s + (−2.10 + 2.56i)11-s + (1.66 − 0.963i)12-s + 5.43·13-s + (−2.86 − 4.39i)14-s + 0.364·15-s + (2.07 + 3.58i)16-s + (−2.93 + 5.08i)17-s + ⋯
L(s)  = 1  + (−1.21 + 0.700i)2-s + (0.499 + 0.288i)3-s + (0.481 − 0.834i)4-s + (0.141 − 0.0815i)5-s − 0.808·6-s + (0.0541 + 0.998i)7-s − 0.0512i·8-s + (0.166 + 0.288i)9-s + (−0.114 + 0.197i)10-s + (−0.634 + 0.773i)11-s + (0.481 − 0.278i)12-s + 1.50·13-s + (−0.765 − 1.17i)14-s + 0.0941·15-s + (0.517 + 0.896i)16-s + (−0.712 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.515 - 0.857i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.515 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369087 + 0.652553i\)
\(L(\frac12)\) \(\approx\) \(0.369087 + 0.652553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.143 - 2.64i)T \)
11 \( 1 + (2.10 - 2.56i)T \)
good2 \( 1 + (1.71 - 0.990i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.315 + 0.182i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 5.43T + 13T^{2} \)
17 \( 1 + (2.93 - 5.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.11 + 1.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.38 + 2.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (4.99 + 2.88i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.00 - 6.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.88T + 41T^{2} \)
43 \( 1 + 5.90iT - 43T^{2} \)
47 \( 1 + (-5.45 + 3.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.47 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.39 - 4.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.35 - 2.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.30 + 12.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.64T + 71T^{2} \)
73 \( 1 + (-2.91 + 5.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.59 - 0.919i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 + (0.367 - 0.212i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.72372521520698499440711707089, −11.16417952314107648692500370584, −10.28693612927726032034693575095, −9.254053040180778505668856584087, −8.637408421975687704700508314619, −7.943101958180074609124987426213, −6.66230459963405238686549576078, −5.62989084488028580322163436081, −3.92532367843491983947012796890, −2.01581465254908331880533485700, 0.908363137298125175935604763562, 2.51662744591993970586521640173, 3.92920140945777123126008613132, 5.86658925772628152792448695997, 7.32348842840594475248950185168, 8.149985475126289662689679418688, 8.982461785038952363758638044679, 9.919430745464742852020876409364, 10.92501859073471687709138745695, 11.31238048600864981276225307888

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