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Properties

Label 2-5e3-25.9-c1-0-1
Degree $2$
Conductor $125$
Sign $0.770 - 0.637i$
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  + (−0.666 + 0.917i)2-s + (2.47 − 0.804i)3-s + (0.220 + 0.679i)4-s + (−0.911 + 2.80i)6-s − 0.407i·7-s + (−2.92 − 0.951i)8-s + (3.05 − 2.21i)9-s + (−1.61 − 1.17i)11-s + (1.09 + 1.50i)12-s + (0.411 + 0.566i)13-s + (0.373 + 0.271i)14-s + (1.66 − 1.21i)16-s + (−1.50 − 0.489i)17-s + 4.27i·18-s + (−1.52 + 4.70i)19-s + ⋯
L(s)  = 1  + (−0.471 + 0.648i)2-s + (1.42 − 0.464i)3-s + (0.110 + 0.339i)4-s + (−0.372 + 1.14i)6-s − 0.153i·7-s + (−1.03 − 0.336i)8-s + (1.01 − 0.739i)9-s + (−0.487 − 0.354i)11-s + (0.315 + 0.434i)12-s + (0.114 + 0.157i)13-s + (0.0998 + 0.0725i)14-s + (0.416 − 0.302i)16-s + (−0.365 − 0.118i)17-s + 1.00i·18-s + (−0.350 + 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16794 + 0.420817i\)
\(L(\frac12)\) \(\approx\) \(1.16794 + 0.420817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.666 - 0.917i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.47 + 0.804i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 0.407iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.411 - 0.566i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.50 + 0.489i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.52 - 4.70i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.706 + 0.971i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.70 + 5.23i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.53 + 7.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.01 + 4.15i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.83 - 4.24i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 + (1.21 - 0.393i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.83 + 1.56i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.25 - 3.82i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.62 - 5.53i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.93 - 0.952i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.12 - 6.53i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.320 + 0.441i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.69 + 5.21i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.926 - 0.301i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.83 + 1.33i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-14.4 + 4.70i)T + (78.4 - 57.0i)T^{2} \)
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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

−13.53234751803005608609734622712, −12.84121511724302375801694463765, −11.58078248634833501648502420527, −9.943513593663238296594697870249, −8.855375715952381064152512083647, −8.107657744509691137420166258446, −7.41603576166289918174158650704, −6.15209028999264814683207120899, −3.82341485813312170194172909978, −2.49237423625452861861361653763, 2.15338723956523900867783125274, 3.31707027969180919959931698209, 5.08694410906755579888508315051, 6.94025423303678543366079538396, 8.505947718401594819023955054318, 9.036179740428694165322432625339, 10.10271382905173338720855172799, 10.83803102023852759800109745027, 12.19670172901575572313907518503, 13.42688539338879639062757681476

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