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Properties

Label 2-725-145.139-c1-0-13
Degree $2$
Conductor $725$
Sign $-0.233 + 0.972i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
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  + (−2.18 − 1.74i)2-s + (−0.690 − 0.157i)3-s + (1.30 + 5.69i)4-s + (1.23 + 1.55i)6-s + (−1.77 − 0.404i)7-s + (4.67 − 9.70i)8-s + (−2.25 − 1.08i)9-s + (3.09 − 1.49i)11-s − 4.14i·12-s + (0.547 + 1.13i)13-s + (3.17 + 3.98i)14-s + (−16.6 + 8.01i)16-s + 6.04i·17-s + (3.03 + 6.30i)18-s + (0.717 + 3.14i)19-s + ⋯
L(s)  = 1  + (−1.54 − 1.23i)2-s + (−0.398 − 0.0910i)3-s + (0.650 + 2.84i)4-s + (0.505 + 0.633i)6-s + (−0.669 − 0.152i)7-s + (1.65 − 3.43i)8-s + (−0.750 − 0.361i)9-s + (0.934 − 0.450i)11-s − 1.19i·12-s + (0.151 + 0.315i)13-s + (0.848 + 1.06i)14-s + (−4.16 + 2.00i)16-s + 1.46i·17-s + (0.715 + 1.48i)18-s + (0.164 + 0.721i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.233 + 0.972i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.233 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.294509 - 0.373708i\)
\(L(\frac12)\) \(\approx\) \(0.294509 - 0.373708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (-0.718 + 5.33i)T \)
good2 \( 1 + (2.18 + 1.74i)T + (0.445 + 1.94i)T^{2} \)
3 \( 1 + (0.690 + 0.157i)T + (2.70 + 1.30i)T^{2} \)
7 \( 1 + (1.77 + 0.404i)T + (6.30 + 3.03i)T^{2} \)
11 \( 1 + (-3.09 + 1.49i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.547 - 1.13i)T + (-8.10 + 10.1i)T^{2} \)
17 \( 1 - 6.04iT - 17T^{2} \)
19 \( 1 + (-0.717 - 3.14i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-0.539 + 0.429i)T + (5.11 - 22.4i)T^{2} \)
31 \( 1 + (-0.304 + 0.381i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (-2.34 + 4.86i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 3.37T + 41T^{2} \)
43 \( 1 + (2.28 - 1.82i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (3.79 + 7.87i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (-4.30 - 3.43i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + (-1.82 + 8.00i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-6.25 + 12.9i)T + (-41.7 - 52.3i)T^{2} \)
71 \( 1 + (-3.47 + 1.67i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.92 + 1.53i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-0.240 - 0.116i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (-5.40 + 1.23i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.696 + 0.873i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-1.10 + 0.252i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.10041749637365708738733102933, −9.444895917347804924498972519484, −8.632082205923462162251128018250, −8.027990109892450751155301699871, −6.75985628364242748294996047795, −6.12286802364928211555022204708, −3.94755804285762696084662605528, −3.36280338331532377779921980655, −1.95594622395093672968547111726, −0.61401716734494141071712779465, 0.891000074695386284543666184734, 2.63473501439399953237547457102, 4.86537993293194456463317094079, 5.59829446151895951186383406397, 6.60278969479061496000000984407, 7.05279129527233464899792469090, 8.123172269517392413185846324449, 8.986250627982951147900783524394, 9.513965950451171135890239763189, 10.28256826777032957585564484920

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