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Properties

Label 2-5e3-5.4-c1-0-5
Degree $2$
Conductor $125$
Sign $i$
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  − 1.61i·2-s + 0.381i·3-s − 0.618·4-s + 0.618·6-s − 3i·7-s − 2.23i·8-s + 2.85·9-s − 3·11-s − 0.236i·12-s + 4.85i·13-s − 4.85·14-s − 4.85·16-s + 4.23i·17-s − 4.61i·18-s + 3.61·19-s + ⋯
L(s)  = 1  − 1.14i·2-s + 0.220i·3-s − 0.309·4-s + 0.252·6-s − 1.13i·7-s − 0.790i·8-s + 0.951·9-s − 0.904·11-s − 0.0681i·12-s + 1.34i·13-s − 1.29·14-s − 1.21·16-s + 1.02i·17-s − 1.08i·18-s + 0.830·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.809179 - 0.809179i\)
\(L(\frac12)\) \(\approx\) \(0.809179 - 0.809179i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61iT - 2T^{2} \)
3 \( 1 - 0.381iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4.85iT - 13T^{2} \)
17 \( 1 - 4.23iT - 17T^{2} \)
19 \( 1 - 3.61T + 19T^{2} \)
23 \( 1 - 1.23iT - 23T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 - 3.70iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 + 8.32iT - 47T^{2} \)
53 \( 1 + 4.61iT - 53T^{2} \)
59 \( 1 + 4.14T + 59T^{2} \)
61 \( 1 + 6.09T + 61T^{2} \)
67 \( 1 + 13.8iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 1.85iT - 73T^{2} \)
79 \( 1 + 0.527T + 79T^{2} \)
83 \( 1 + 0.472iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 7.85iT - 97T^{2} \)
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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

−13.11068815054791527609486374286, −11.97634633614673577383643678829, −10.96180908249629558499458263771, −10.22559838753894048930523142459, −9.464004666535762095230076097221, −7.67934077724085692414553132739, −6.66093478675703668570081715564, −4.56692500945719828149017556503, −3.56268744236630577944616204915, −1.63766590296134770880063997587, 2.64404856743898449754712236702, 5.08160941167737285257424150641, 5.84127573365123529794145698744, 7.26286205717751339556100128562, 7.932819063088943909246232931950, 9.161775503735509442040481260677, 10.45119452076391228146257628859, 11.77715468349145611881824683349, 12.77956334825184657944251723542, 13.75475604773128100308064988413

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