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Properties

Label 2-5e3-25.19-c1-0-1
Degree $2$
Conductor $125$
Sign $0.436 + 0.899i$
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  + (−1.53 + 0.5i)2-s + (−0.587 − 0.809i)3-s + (0.5 − 0.363i)4-s + (1.30 + 0.951i)6-s − 0.618i·7-s + (1.31 − 1.80i)8-s + (0.618 − 1.90i)9-s + (−1.61 − 4.97i)11-s + (−0.587 − 0.190i)12-s + (1.76 + 0.572i)13-s + (0.309 + 0.951i)14-s + (−1.50 + 4.61i)16-s + (3.07 − 4.23i)17-s + 3.23i·18-s + (0.690 + 0.502i)19-s + ⋯
L(s)  = 1  + (−1.08 + 0.353i)2-s + (−0.339 − 0.467i)3-s + (0.250 − 0.181i)4-s + (0.534 + 0.388i)6-s − 0.233i·7-s + (0.464 − 0.639i)8-s + (0.206 − 0.634i)9-s + (−0.487 − 1.50i)11-s + (−0.169 − 0.0551i)12-s + (0.489 + 0.158i)13-s + (0.0825 + 0.254i)14-s + (−0.375 + 1.15i)16-s + (0.746 − 1.02i)17-s + 0.762i·18-s + (0.158 + 0.115i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.436 + 0.899i$
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: \((2,\ 125,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421592 - 0.264041i\)
\(L(\frac12)\) \(\approx\) \(0.421592 - 0.264041i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 0.5i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.587 + 0.809i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.618iT - 7T^{2} \)
11 \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.76 - 0.572i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.07 + 4.23i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.690 - 0.502i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.57 - 1.16i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.92 - 2.12i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.42 - 1.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.224 - 0.0729i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.236 - 0.726i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.85iT - 43T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.04 - 2.80i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.35 + 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.69 - 8.28i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.80 - 3.85i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-5.35 + 3.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.55 + 2.78i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.54 - 4.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.66 - 5.04i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.76 - 8.50i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.26 + 3.11i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.27783753121658280529549743903, −12.07060420389777768739380127348, −11.00466496761026649724976613397, −9.893628305948710679568794857429, −8.874270316622056629678748612294, −7.88236412906786309843159776419, −6.88441802301984544498556194121, −5.69746366203112538529994426349, −3.62812310243019469106802979994, −0.835682917278180444937163876946, 2.00941807045984137695360907115, 4.40696189794753581492843109110, 5.62349778898807181124067738562, 7.49592973576757152531307133444, 8.351649119374772113152207343409, 9.752493802480971913766699199979, 10.20760026087833821465771107326, 11.13224830495819489446084991521, 12.32403665881400707431415693133, 13.46150816842799056782442531005

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