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Properties

Label 2-5e3-5.4-c1-0-3
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  + 0.618i·2-s + 2.61i·3-s + 1.61·4-s − 1.61·6-s − 3i·7-s + 2.23i·8-s − 3.85·9-s − 3·11-s + 4.23i·12-s − 1.85i·13-s + 1.85·14-s + 1.85·16-s − 0.236i·17-s − 2.38i·18-s + 1.38·19-s + ⋯
L(s)  = 1  + 0.437i·2-s + 1.51i·3-s + 0.809·4-s − 0.660·6-s − 1.13i·7-s + 0.790i·8-s − 1.28·9-s − 0.904·11-s + 1.22i·12-s − 0.514i·13-s + 0.495·14-s + 0.463·16-s − 0.0572i·17-s − 0.561i·18-s + 0.317·19-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

\(L(1)\) \(\approx\) \(0.857475 + 0.857475i\)
\(L(\frac12)\) \(\approx\) \(0.857475 + 0.857475i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.618iT - 2T^{2} \)
3 \( 1 - 2.61iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 1.85iT - 13T^{2} \)
17 \( 1 + 0.236iT - 17T^{2} \)
19 \( 1 - 1.38T + 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 9.70iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 9iT - 43T^{2} \)
47 \( 1 - 7.32iT - 47T^{2} \)
53 \( 1 + 2.38iT - 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 5.09T + 61T^{2} \)
67 \( 1 + 7.14iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 4.85iT - 73T^{2} \)
79 \( 1 + 9.47T + 79T^{2} \)
83 \( 1 - 8.47iT - 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 1.14iT - 97T^{2} \)
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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

−14.04871600801560644843878857996, −12.62061451744776351396979523276, −11.02272002138538434128028378598, −10.66013892724181831462783203186, −9.734724301156102585413699009746, −8.230380269465210665173673383267, −7.17477285549925130664772821240, −5.67584271451411409228516651073, −4.51300228381581014856939607308, −3.07255736305162712890079456128, 1.78640108322030520370880842804, 2.87693696820391437921529684968, 5.55657437510367856356850031625, 6.63844559564860803910574345364, 7.56736307971967879996830205304, 8.677914879967749046938520999121, 10.20793541307061090781141009377, 11.56937028705761659135978944240, 12.06383036573245201175780565553, 12.88523601482124869524747488285

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