↑

Properties

Label 2-5e3-25.9-c1-0-2
Degree $2$
Conductor $125$
Sign $0.979 + 0.199i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
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  + (0.363 − 0.5i)2-s + (0.951 − 0.309i)3-s + (0.5 + 1.53i)4-s + (0.190 − 0.587i)6-s − 1.61i·7-s + (2.12 + 0.690i)8-s + (−1.61 + 1.17i)9-s + (0.618 + 0.449i)11-s + (0.951 + 1.30i)12-s + (−2.85 − 3.92i)13-s + (−0.809 − 0.587i)14-s + (−1.49 + 1.08i)16-s + (−0.726 − 0.236i)17-s + 1.23i·18-s + (1.80 − 5.56i)19-s + ⋯
L(s)  = 1  + (0.256 − 0.353i)2-s + (0.549 − 0.178i)3-s + (0.250 + 0.769i)4-s + (0.0779 − 0.239i)6-s − 0.611i·7-s + (0.751 + 0.244i)8-s + (−0.539 + 0.391i)9-s + (0.186 + 0.135i)11-s + (0.274 + 0.377i)12-s + (−0.791 − 1.08i)13-s + (−0.216 − 0.157i)14-s + (−0.374 + 0.272i)16-s + (−0.176 − 0.0572i)17-s + 0.291i·18-s + (0.415 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.979 + 0.199i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.979 + 0.199i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40697 - 0.141457i\)
\(L(\frac12)\) \(\approx\) \(1.40697 - 0.141457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.363 + 0.5i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.951 + 0.309i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 1.61iT - 7T^{2} \)
11 \( 1 + (-0.618 - 0.449i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.85 + 3.92i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.726 + 0.236i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.80 + 5.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.84 - 6.66i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.427 - 1.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.927 - 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.48 + 3.42i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.23 + 3.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.85iT - 43T^{2} \)
47 \( 1 + (-1.53 + 0.5i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.20 + 1.69i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.35 - 2.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.80 - 2.76i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-8.78 - 2.85i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.35 + 4.16i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.29 + 7.28i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.954 + 2.93i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.67 - 0.545i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.23 - 5.25i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.71 - 0.881i)T + (78.4 - 57.0i)T^{2} \)
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

−13.42996889224292319738689593528, −12.41861077670510184174803919473, −11.41260422153206943876439599616, −10.44022063963228711340226261673, −9.031646284280891311671038748944, −7.81288711844399261343355314359, −7.20865851972886647019217284652, −5.22098847699734075724587040962, −3.65572391555640403235721900256, −2.45431275345604760836139367433, 2.28492847914089115907774365533, 4.21038082734385381900307654475, 5.69239703882954894757895829388, 6.62001427109455670798930831976, 8.109034188027527382939661238676, 9.282475030142476382446526188239, 10.09612412131775675683295721653, 11.48852765489448724418791556620, 12.33859829840592376140034561383, 13.96867489831347886591319102273

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