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Properties

Label 2-5e3-25.21-c1-0-1
Degree $2$
Conductor $125$
Sign $0.727 - 0.686i$
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  + (−0.0566 + 0.174i)2-s + (1.19 + 0.865i)3-s + (1.59 + 1.15i)4-s + (−0.218 + 0.158i)6-s − 3.26·7-s + (−0.587 + 0.427i)8-s + (−0.257 − 0.792i)9-s + (0.618 − 1.90i)11-s + (0.894 + 2.75i)12-s + (−0.0915 − 0.281i)13-s + (0.184 − 0.568i)14-s + (1.17 + 3.61i)16-s + (4.17 − 3.03i)17-s + 0.152·18-s + (−1.39 + 1.01i)19-s + ⋯
L(s)  = 1  + (−0.0400 + 0.123i)2-s + (0.687 + 0.499i)3-s + (0.795 + 0.577i)4-s + (−0.0890 + 0.0646i)6-s − 1.23·7-s + (−0.207 + 0.150i)8-s + (−0.0858 − 0.264i)9-s + (0.186 − 0.573i)11-s + (0.258 + 0.794i)12-s + (−0.0254 − 0.0781i)13-s + (0.0493 − 0.151i)14-s + (0.293 + 0.903i)16-s + (1.01 − 0.736i)17-s + 0.0359·18-s + (−0.321 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.727 - 0.686i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22665 + 0.487620i\)
\(L(\frac12)\) \(\approx\) \(1.22665 + 0.487620i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0566 - 0.174i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.19 - 0.865i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 3.26T + 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.0915 + 0.281i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.17 + 3.03i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.39 - 1.01i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.271 + 0.836i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.78 + 3.47i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.49 - 7.69i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.313 + 0.965i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + (3.41 + 2.48i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.55 - 4.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.83 - 5.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.282 + 0.870i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.57 - 4.04i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.82 - 3.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.72 + 8.40i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.27 - 4.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.7 - 8.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.32 + 7.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.44 - 3.95i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.52016358084795300957598612868, −12.46658062554058392586285566707, −11.57959825945343335194804780975, −10.21971939625559442113787278125, −9.296494889021379256854262362492, −8.246101076631870883716481182320, −6.98452491268638265727893106094, −5.92081396376645773765390718137, −3.68287449583831048021702364936, −2.91706325090852802275425640592, 1.99860633584028697435867361748, 3.39363909801749772319279957507, 5.62307170451128624244801836727, 6.79509748218383325942098238751, 7.69123170681551984270236107475, 9.196223127964169654533174012535, 10.08906771273092520357681445819, 11.15006608855088464412009226067, 12.48903598887173356942931430668, 13.10962338469966004222464178383

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