Properties

Label 2-5e3-125.11-c1-0-4
Degree $2$
Conductor $125$
Sign $0.758 - 0.651i$
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  + (−1.97 + 1.85i)2-s + (3.22e−5 + 5.08e−5i)3-s + (0.333 − 5.30i)4-s + (2.21 − 0.321i)5-s + (−0.000157 − 4.05e−5i)6-s + (0.678 − 2.08i)7-s + (5.71 + 6.91i)8-s + (1.27 − 2.71i)9-s + (−3.76 + 4.73i)10-s + (−0.314 + 0.295i)11-s + (0.000280 − 0.000154i)12-s + (−0.700 + 1.48i)13-s + (2.52 + 5.37i)14-s + (8.78e−5 + 0.000102i)15-s + (−13.5 − 1.70i)16-s + (0.391 + 6.22i)17-s + ⋯
L(s)  = 1  + (−1.39 + 1.30i)2-s + (1.86e−5 + 2.93e−5i)3-s + (0.166 − 2.65i)4-s + (0.989 − 0.143i)5-s + (−6.44e−5 − 1.65e−5i)6-s + (0.256 − 0.789i)7-s + (2.02 + 2.44i)8-s + (0.425 − 0.904i)9-s + (−1.19 + 1.49i)10-s + (−0.0949 + 0.0891i)11-s + (8.10e−5 − 4.45e−5i)12-s + (−0.194 + 0.412i)13-s + (0.676 + 1.43i)14-s + (2.26e−5 + 2.63e−5i)15-s + (−3.38 − 0.427i)16-s + (0.0949 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.622535 + 0.230721i\)
\(L(\frac12)\) \(\approx\) \(0.622535 + 0.230721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 + 0.321i)T \)
good2 \( 1 + (1.97 - 1.85i)T + (0.125 - 1.99i)T^{2} \)
3 \( 1 + (-3.22e-5 - 5.08e-5i)T + (-1.27 + 2.71i)T^{2} \)
7 \( 1 + (-0.678 + 2.08i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.314 - 0.295i)T + (0.690 - 10.9i)T^{2} \)
13 \( 1 + (0.700 - 1.48i)T + (-8.28 - 10.0i)T^{2} \)
17 \( 1 + (-0.391 - 6.22i)T + (-16.8 + 2.13i)T^{2} \)
19 \( 1 + (-0.817 + 1.28i)T + (-8.08 - 17.1i)T^{2} \)
23 \( 1 + (-1.52 + 7.98i)T + (-21.3 - 8.46i)T^{2} \)
29 \( 1 + (-2.66 - 1.05i)T + (21.1 + 19.8i)T^{2} \)
31 \( 1 + (-0.396 - 6.30i)T + (-30.7 + 3.88i)T^{2} \)
37 \( 1 + (4.09 + 0.516i)T + (35.8 + 9.20i)T^{2} \)
41 \( 1 + (-0.991 - 5.19i)T + (-38.1 + 15.0i)T^{2} \)
43 \( 1 + (8.79 + 6.38i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (4.96 - 5.99i)T + (-8.80 - 46.1i)T^{2} \)
53 \( 1 + (6.71 - 1.72i)T + (46.4 - 25.5i)T^{2} \)
59 \( 1 + (7.99 - 4.39i)T + (31.6 - 49.8i)T^{2} \)
61 \( 1 + (-1.00 + 5.26i)T + (-56.7 - 22.4i)T^{2} \)
67 \( 1 + (-6.20 + 2.45i)T + (48.8 - 45.8i)T^{2} \)
71 \( 1 + (-0.931 + 1.12i)T + (-13.3 - 69.7i)T^{2} \)
73 \( 1 + (-4.80 - 2.64i)T + (39.1 + 61.6i)T^{2} \)
79 \( 1 + (1.18 + 1.86i)T + (-33.6 + 71.4i)T^{2} \)
83 \( 1 + (3.90 - 6.14i)T + (-35.3 - 75.1i)T^{2} \)
89 \( 1 + (8.79 + 4.83i)T + (47.6 + 75.1i)T^{2} \)
97 \( 1 + (-3.97 - 1.57i)T + (70.7 + 66.4i)T^{2} \)
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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

−14.05387351341722934342605684340, −12.65118849796019261938262037046, −10.69575347909375199214708522319, −10.13964654347103945884552394122, −9.137587249859081401435646679495, −8.262412124606267674393196178171, −6.85078046509076994393327457878, −6.33159279468352346243604103199, −4.81769138024819847178453366485, −1.38037401506999444852499718237, 1.81118916353498624844010760839, 2.95184325961224702396788513203, 5.20240990725953627775451036833, 7.24293663969163400059332753132, 8.277213692672853454601329516405, 9.478512358113120877237391224505, 9.953746530736133283274704033714, 11.09236911244840407562347940129, 11.86987978967350805528486054626, 13.03878049876593424294480684901

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