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Properties

Label 2-5e3-25.9-c1-0-0
Degree $2$
Conductor $125$
Sign $-0.260 - 0.965i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.260 - 0.965i$
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: \((2,\ 125,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479492 + 0.625819i\)
\(L(\frac12)\) \(\approx\) \(0.479492 + 0.625819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.71 + 0.881i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.63974810285739358097188022684, −12.48067015661411841525621784400, −11.59132628020091659413299466820, −10.88813043030972794353522054889, −9.133398696760145106091004822391, −8.525745837847073596760729010942, −7.06781254693130290727211041631, −6.10696930285051884067001288940, −4.61827464846276781627458285704, −2.80177642705823829414931113637, 1.06139388462430419569641658532, 3.37012427512599979918941162197, 5.46949551689546964019650255098, 6.18292146690344596959202432049, 7.64122396641957617062150311220, 9.088041240447273545525438254320, 10.17982921280987079962729677283, 11.05541988196755737550548724818, 11.75636211985799801969392413784, 12.99476048181578427342208010324

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