Properties

Label 2-5e3-25.4-c1-0-2
Degree $2$
Conductor $125$
Sign $0.999 - 0.0119i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.999 - 0.0119i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.999 - 0.0119i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75107 + 0.0104297i\)
\(L(\frac12)\) \(\approx\) \(1.75107 + 0.0104297i\)
\(L(\frac{3}{2})\) 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

−13.38655214117219619583524614566, −12.81940785564940309183775353282, −11.76190255681960971235057140420, −10.24547010934012805961462139104, −9.207899202156275079575525538728, −7.48929204040530819302803307932, −6.92760839541108900319346148602, −5.21327529208244492984302149512, −4.39831789353184290364788653501, −2.48835521061366510424077816978, 2.88568655688020667946843684064, 3.89979255863015220012740843026, 5.21613428887431790206537603324, 6.36851463574156010314540832057, 8.292355546489783698757089201217, 9.089053569353877183654968439815, 10.54827177959549959441677451041, 11.49783687581234067281586101309, 12.61210400246864807858921687590, 13.30768359413217824033643682202

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