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Properties

Label 2-5e3-25.19-c1-0-0
Degree $2$
Conductor $125$
Sign $0.201 - 0.979i$
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  + (−0.174 + 0.0566i)2-s + (0.865 + 1.19i)3-s + (−1.59 + 1.15i)4-s + (−0.218 − 0.158i)6-s + 3.26i·7-s + (0.427 − 0.587i)8-s + (0.257 − 0.792i)9-s + (0.618 + 1.90i)11-s + (−2.75 − 0.894i)12-s + (−0.281 − 0.0915i)13-s + (−0.184 − 0.568i)14-s + (1.17 − 3.61i)16-s + (3.03 − 4.17i)17-s + 0.152i·18-s + (1.39 + 1.01i)19-s + ⋯
L(s)  = 1  + (−0.123 + 0.0400i)2-s + (0.499 + 0.687i)3-s + (−0.795 + 0.577i)4-s + (−0.0890 − 0.0646i)6-s + 1.23i·7-s + (0.150 − 0.207i)8-s + (0.0858 − 0.264i)9-s + (0.186 + 0.573i)11-s + (−0.794 − 0.258i)12-s + (−0.0781 − 0.0254i)13-s + (−0.0493 − 0.151i)14-s + (0.293 − 0.903i)16-s + (0.736 − 1.01i)17-s + 0.0359i·18-s + (0.321 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.201 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781879 + 0.637228i\)
\(L(\frac12)\) \(\approx\) \(0.781879 + 0.637228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.174 - 0.0566i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.865 - 1.19i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 3.26iT - 7T^{2} \)
11 \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.281 + 0.0915i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.03 + 4.17i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.39 - 1.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.836 - 0.271i)T + (18.6 - 13.5i)T^{2} \)
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 \( 1 + (7.69 + 2.49i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.313 - 0.965i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.24iT - 43T^{2} \)
47 \( 1 + (-2.48 - 3.41i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-4.76 - 6.55i)T + (-16.3 + 50.4i)T^{2} \)
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 \( 1 + (4.04 - 5.57i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.82 + 3.50i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.40 - 2.72i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.27 - 4.56i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.53 + 11.7i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.32 + 7.15i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (3.95 + 5.44i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.75984406700575009318268963387, −12.38770440841059662788359765610, −11.93651098008618023452115403241, −10.05118257872306870574534744214, −9.310774601339802778046175499673, −8.634158962708150822044236508669, −7.35971081650753578013169725912, −5.52733087696146860919872411262, −4.26080338892158134598736205753, −2.94400525068879315617137458881, 1.33159854890617141344374262893, 3.68454307063217068985201876558, 5.15115860058602761065731726403, 6.72584172992979940184138309833, 7.893192031649252102695770862246, 8.775734038678288717871432347072, 10.16877243344851377443186496315, 10.78029948018017248071307202476, 12.45770166895612968314388744428, 13.45609356681184142869936693093

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