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Properties

Label 2-5e4-25.6-c1-0-6
Degree $2$
Conductor $625$
Sign $-0.783 - 0.621i$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
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  + (0.520 + 1.60i)2-s + (0.574 − 0.417i)3-s + (−0.674 + 0.490i)4-s + (0.967 + 0.702i)6-s − 4.59·7-s + (1.58 + 1.15i)8-s + (−0.771 + 2.37i)9-s + (1.20 + 3.72i)11-s + (−0.183 + 0.563i)12-s + (−0.177 + 0.544i)13-s + (−2.38 − 7.35i)14-s + (−1.53 + 4.72i)16-s + (0.188 + 0.136i)17-s − 4.20·18-s + (4.49 + 3.26i)19-s + ⋯
L(s)  = 1  + (0.367 + 1.13i)2-s + (0.331 − 0.241i)3-s + (−0.337 + 0.245i)4-s + (0.394 + 0.286i)6-s − 1.73·7-s + (0.561 + 0.407i)8-s + (−0.257 + 0.791i)9-s + (0.364 + 1.12i)11-s + (−0.0528 + 0.162i)12-s + (−0.0491 + 0.151i)13-s + (−0.638 − 1.96i)14-s + (−0.384 + 1.18i)16-s + (0.0456 + 0.0331i)17-s − 0.990·18-s + (1.03 + 0.748i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-0.783 - 0.621i$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{625} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538693 + 1.54691i\)
\(L(\frac12)\) \(\approx\) \(0.538693 + 1.54691i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.520 - 1.60i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.574 + 0.417i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 4.59T + 7T^{2} \)
11 \( 1 + (-1.20 - 3.72i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.177 - 0.544i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.188 - 0.136i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.49 - 3.26i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.52 + 4.69i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.34 - 2.42i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.82 - 2.05i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.67 - 5.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.22 + 9.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + (0.744 - 0.541i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.996 - 0.723i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.39 + 4.28i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.60 + 11.0i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-2.39 - 1.73i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (2.59 - 1.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.18 + 9.78i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.77 + 5.65i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.48 - 6.16i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.24 - 6.90i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.73 + 4.89i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.73247462627147511129773240413, −9.960543496306233940720082025011, −9.105683619009540423483471019315, −8.007119519194936558477422798762, −7.19734687804620979746338129954, −6.60444274146366518380569633614, −5.71611897800616749512773205471, −4.69221916010842097915620840798, −3.42104115049825378193118407511, −2.06970167329321192316007237270, 0.76145319698872502959377538504, 2.78479552189477911817251514721, 3.32179746104676770632426181037, 4.03270158893829285209459470195, 5.70646493992342041766409811448, 6.53801713341943072376854950586, 7.58862726000448936101453207176, 9.051098797123298057834187809356, 9.528038513999281807549235205301, 10.21006353562534337121339670717

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