Properties

Label 2-5e3-25.11-c1-0-2
Degree $2$
Conductor $125$
Sign $0.728 - 0.684i$
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  + (0.5 − 0.363i)2-s + (−0.309 + 0.951i)3-s + (−0.5 + 1.53i)4-s + (0.190 + 0.587i)6-s + 1.61·7-s + (0.690 + 2.12i)8-s + (1.61 + 1.17i)9-s + (0.618 − 0.449i)11-s + (−1.30 − 0.951i)12-s + (−3.92 − 2.85i)13-s + (0.809 − 0.587i)14-s + (−1.49 − 1.08i)16-s + (0.236 + 0.726i)17-s + 1.23·18-s + (−1.80 − 5.56i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.256i)2-s + (−0.178 + 0.549i)3-s + (−0.250 + 0.769i)4-s + (0.0779 + 0.239i)6-s + 0.611·7-s + (0.244 + 0.751i)8-s + (0.539 + 0.391i)9-s + (0.186 − 0.135i)11-s + (−0.377 − 0.274i)12-s + (−1.08 − 0.791i)13-s + (0.216 − 0.157i)14-s + (−0.374 − 0.272i)16-s + (0.0572 + 0.176i)17-s + 0.291·18-s + (−0.415 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.728 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11556 + 0.441684i\)
\(L(\frac12)\) \(\approx\) \(1.11556 + 0.441684i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.363i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.309 - 0.951i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.92 + 2.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.236 - 0.726i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.66 + 4.84i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.427 - 1.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.42 - 2.48i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (-0.5 + 1.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.69 - 5.20i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.35 - 2.43i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.80 + 2.76i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.85 + 8.78i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.35 - 4.16i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.28 - 5.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.954 + 2.93i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.545 - 1.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.23 - 5.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.881 - 2.71i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.23518809896114327303127719159, −12.65789668423712064696770041157, −11.41091436612210380689717288077, −10.65589798377282448672873700748, −9.344765813345424687886412713960, −8.169094411065815224186471539512, −7.13074336729048113073877128460, −5.08641636657488064733419266759, −4.41118346850708511770952688303, −2.72787109694886022922365888462, 1.58565721465164041901166539504, 4.20400976548610655520605483032, 5.37505238086176902259383955208, 6.64732879829933377158895834460, 7.53106459787801988135621123294, 9.199137839270798980881721888229, 10.06250877612428110861837420383, 11.36653328904768638982232144502, 12.39550509346548064122666511511, 13.30291166082849356869043968307

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