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Properties

Label 2-5e3-25.6-c1-0-0
Degree $2$
Conductor $125$
Sign $-0.974 - 0.222i$
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  + (0.644 + 1.98i)2-s + (−1.77 + 1.29i)3-s + (−1.89 + 1.38i)4-s + (−3.70 − 2.69i)6-s − 0.992·7-s + (−0.587 − 0.427i)8-s + (0.566 − 1.74i)9-s + (0.618 + 1.90i)11-s + (1.59 − 4.91i)12-s + (1.04 − 3.20i)13-s + (−0.639 − 1.96i)14-s + (−0.983 + 3.02i)16-s + (2.34 + 1.70i)17-s + 3.82·18-s + (2.09 + 1.51i)19-s + ⋯
L(s)  = 1  + (0.455 + 1.40i)2-s + (−1.02 + 0.746i)3-s + (−0.949 + 0.690i)4-s + (−1.51 − 1.10i)6-s − 0.375·7-s + (−0.207 − 0.150i)8-s + (0.188 − 0.581i)9-s + (0.186 + 0.573i)11-s + (0.460 − 1.41i)12-s + (0.289 − 0.889i)13-s + (−0.170 − 0.526i)14-s + (−0.245 + 0.756i)16-s + (0.567 + 0.412i)17-s + 0.901·18-s + (0.479 + 0.348i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.974 - 0.222i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ -0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106157 + 0.940684i\)
\(L(\frac12)\) \(\approx\) \(0.106157 + 0.940684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.644 - 1.98i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.77 - 1.29i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.992T + 7T^{2} \)
11 \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.04 + 3.20i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.34 - 1.70i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.09 - 1.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.40 - 4.32i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.35 + 3.16i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.110 + 0.0802i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.664 + 2.04i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.66 + 8.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.64T + 43T^{2} \)
47 \( 1 + (-8.03 + 5.83i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-6.12 + 4.44i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.51 + 4.67i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.855 + 2.63i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.76 - 1.28i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (7.80 - 5.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.239 + 0.737i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.8 - 9.31i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.43 + 1.04i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.48 - 13.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (13.7 - 10.0i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.10959918910686807605753565375, −13.04104771635002301526338403574, −11.90983684761856381311757061035, −10.68009400425123140966934183705, −9.778178490966372419216154112622, −8.223344224045782344009580613699, −7.06907241426257062043353459666, −5.85616938902760109613537877075, −5.25385183935734417385814863234, −3.93265429496862221039428894318, 1.12639188219607028131991126692, 3.04573600228044058396216539525, 4.65293761762137282576417576334, 6.07899320678413094032656372611, 7.16014306199789855358376156182, 9.024309862091834607697118087478, 10.25669145670302876882142343774, 11.29418684727369131528264229010, 11.82117192301370845144838396486, 12.63295413301885882287911675742

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