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Properties

Label 2-5e3-5.4-c3-0-8
Degree $2$
Conductor $125$
Sign $-i$
Analytic cond. $7.37523$
Root an. cond. $2.71573$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.288i·2-s + 4.57i·3-s + 7.91·4-s − 1.32·6-s + 18.9i·7-s + 4.59i·8-s + 6.03·9-s − 7.94·11-s + 36.2i·12-s − 43.9i·13-s − 5.45·14-s + 62.0·16-s + 124. i·17-s + 1.74i·18-s − 93.4·19-s + ⋯
L(s)  = 1  + 0.102i·2-s + 0.881i·3-s + 0.989·4-s − 0.0899·6-s + 1.02i·7-s + 0.203i·8-s + 0.223·9-s − 0.217·11-s + 0.872i·12-s − 0.937i·13-s − 0.104·14-s + 0.968·16-s + 1.78i·17-s + 0.0228i·18-s − 1.12·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-i$
Analytic conductor: \(7.37523\)
Root analytic conductor: \(2.71573\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :3/2),\ -i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.40250 + 1.40250i\)
\(L(\frac12)\) \(\approx\) \(1.40250 + 1.40250i\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.288iT - 8T^{2} \)
3 \( 1 - 4.57iT - 27T^{2} \)
7 \( 1 - 18.9iT - 343T^{2} \)
11 \( 1 + 7.94T + 1.33e3T^{2} \)
13 \( 1 + 43.9iT - 2.19e3T^{2} \)
17 \( 1 - 124. iT - 4.91e3T^{2} \)
19 \( 1 + 93.4T + 6.85e3T^{2} \)
23 \( 1 - 23.6iT - 1.21e4T^{2} \)
29 \( 1 - 171.T + 2.43e4T^{2} \)
31 \( 1 + 70.6T + 2.97e4T^{2} \)
37 \( 1 + 275. iT - 5.06e4T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 + 433. iT - 7.95e4T^{2} \)
47 \( 1 + 249. iT - 1.03e5T^{2} \)
53 \( 1 + 112. iT - 1.48e5T^{2} \)
59 \( 1 + 844.T + 2.05e5T^{2} \)
61 \( 1 - 768.T + 2.26e5T^{2} \)
67 \( 1 + 385. iT - 3.00e5T^{2} \)
71 \( 1 - 350.T + 3.57e5T^{2} \)
73 \( 1 + 773. iT - 3.89e5T^{2} \)
79 \( 1 - 497.T + 4.93e5T^{2} \)
83 \( 1 + 349. iT - 5.71e5T^{2} \)
89 \( 1 - 1.37e3T + 7.04e5T^{2} \)
97 \( 1 - 1.20e3iT - 9.12e5T^{2} \)
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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

−12.84771485494145944284029981115, −12.21835338814772797366874207427, −10.73220969992105415088007354094, −10.42049033681693588897480140750, −8.935469045805286565294227186854, −7.87860728417648681055428304847, −6.35636716747583122378954092164, −5.37747872086162592840612784149, −3.71468798924064512663684000317, −2.16126628800120035827108369723, 1.10508956714653878920255764714, 2.58113553688683254796617728312, 4.45173609837080375546787349113, 6.46343629144547859038558282442, 7.04575289315191101543489639577, 7.953748805740907917257343792260, 9.670699654537889197781202283499, 10.77453215352747149912349400913, 11.67654340099183304818532071290, 12.61007681632201012994621490136

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