↑

Properties

Label 2-471-157.12-c1-0-25
Degree $2$
Conductor $471$
Sign $-0.699 + 0.714i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
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  + 1.66·2-s + (−0.5 − 0.866i)3-s + 0.770·4-s + (−0.823 − 1.42i)5-s + (−0.832 − 1.44i)6-s − 2.35·7-s − 2.04·8-s + (−0.499 + 0.866i)9-s + (−1.37 − 2.37i)10-s + (−2.90 − 5.02i)11-s + (−0.385 − 0.666i)12-s + (1.56 − 2.70i)13-s − 3.91·14-s + (−0.823 + 1.42i)15-s − 4.94·16-s + (−0.264 − 0.458i)17-s + ⋯
L(s)  = 1  + 1.17·2-s + (−0.288 − 0.499i)3-s + 0.385·4-s + (−0.368 − 0.637i)5-s + (−0.339 − 0.588i)6-s − 0.889·7-s − 0.723·8-s + (−0.166 + 0.288i)9-s + (−0.433 − 0.750i)10-s + (−0.875 − 1.51i)11-s + (−0.111 − 0.192i)12-s + (0.433 − 0.750i)13-s − 1.04·14-s + (−0.212 + 0.368i)15-s − 1.23·16-s + (−0.0641 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.699 + 0.714i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ -0.699 + 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498939 - 1.18744i\)
\(L(\frac12)\) \(\approx\) \(0.498939 - 1.18744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
157 \( 1 + (0.676 - 12.5i)T \)
good2 \( 1 - 1.66T + 2T^{2} \)
5 \( 1 + (0.823 + 1.42i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 + (2.90 + 5.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 2.70i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.264 + 0.458i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.39 - 5.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.18T + 23T^{2} \)
29 \( 1 + 6.84T + 29T^{2} \)
31 \( 1 + (-3.04 + 5.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.506 + 0.876i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.11T + 41T^{2} \)
43 \( 1 + (3.62 - 6.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.90 + 3.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.92 + 8.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + (2.72 + 4.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 5.62T + 67T^{2} \)
71 \( 1 + (2.91 - 5.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.134 + 0.232i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (1.41 - 2.45i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.22 + 12.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.26 - 5.65i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.08272094143279801758529235363, −9.873350916235268350958081573447, −8.680395923351031716877337820556, −7.974030828339079083903414651781, −6.61898267621638041248241321887, −5.67569547391228143718540162205, −5.20538072974742221082710083681, −3.67286561332890889454385641874, −2.98233956704456021316478112653, −0.54931513662725729718353932398, 2.74170074603780186581425269652, 3.57622528094939946957146193714, 4.70643050269062575167963521906, 5.35449824159870149556340100910, 6.79111844136281412813588712795, 7.09575309034157744303112801086, 8.952971708757168976996867671417, 9.604872714495957300162427153662, 10.67510071552109058853794834257, 11.46260393255812884898841323837

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