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Properties

Label 2-5e3-25.4-c1-0-1
Degree $2$
Conductor $125$
Sign $0.322 - 0.946i$
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  + (1.98 + 0.644i)2-s + (−1.29 + 1.77i)3-s + (1.89 + 1.38i)4-s + (−3.70 + 2.69i)6-s − 0.992i·7-s + (0.427 + 0.587i)8-s + (−0.566 − 1.74i)9-s + (0.618 − 1.90i)11-s + (−4.91 + 1.59i)12-s + (3.20 − 1.04i)13-s + (0.639 − 1.96i)14-s + (−0.983 − 3.02i)16-s + (1.70 + 2.34i)17-s − 3.82i·18-s + (−2.09 + 1.51i)19-s + ⋯
L(s)  = 1  + (1.40 + 0.455i)2-s + (−0.746 + 1.02i)3-s + (0.949 + 0.690i)4-s + (−1.51 + 1.10i)6-s − 0.375i·7-s + (0.150 + 0.207i)8-s + (−0.188 − 0.581i)9-s + (0.186 − 0.573i)11-s + (−1.41 + 0.460i)12-s + (0.889 − 0.289i)13-s + (0.170 − 0.526i)14-s + (−0.245 − 0.756i)16-s + (0.412 + 0.567i)17-s − 0.901i·18-s + (−0.479 + 0.348i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.322 - 0.946i$
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: \((2,\ 125,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36204 + 0.974406i\)
\(L(\frac12)\) \(\approx\) \(1.36204 + 0.974406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 - 0.644i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.29 - 1.77i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.992iT - 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-3.20 + 1.04i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.70 - 2.34i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.09 - 1.51i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.32 + 1.40i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.35 + 3.16i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.110 - 0.0802i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.04 - 0.664i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.66 - 8.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.64iT - 43T^{2} \)
47 \( 1 + (5.83 - 8.03i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.44 + 6.12i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.51 + 4.67i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.855 - 2.63i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.28 - 1.76i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (7.80 + 5.66i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.737 - 0.239i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.8 - 9.31i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.04 - 1.43i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.48 - 13.7i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-10.0 + 13.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−13.66584532985350005281119201178, −12.77609754036133549164005695800, −11.60110427131679039285158431459, −10.76233335300863348269522586837, −9.694987187120171287809384051085, −8.030652390428678021163143183820, −6.31273544068046315171361321265, −5.66915721391702084461007968095, −4.40678790073396160840717835838, −3.59406520734639204378027772181, 1.97660272491721220162874227280, 3.82234657133887009705806626236, 5.32768155854376397828334576510, 6.19471456762937751569847836434, 7.26848232502159714176580227826, 8.934116655708991444474141921364, 10.67419970727998753208043652685, 11.76975548141713184359269370666, 12.15277590590322707207563291348, 13.09498240974031742307201393916

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