Properties

Label 2-5e3-125.106-c1-0-8
Degree $2$
Conductor $125$
Sign $0.584 + 0.811i$
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  + (0.399 − 0.630i)2-s + (1.70 − 1.59i)3-s + (0.614 + 1.30i)4-s + (−0.498 − 2.17i)5-s + (−0.326 − 1.71i)6-s + (−3.50 + 2.54i)7-s + (2.54 + 0.322i)8-s + (0.153 − 2.43i)9-s + (−1.57 − 0.557i)10-s + (−2.53 + 3.98i)11-s + (3.13 + 1.23i)12-s + (0.0635 − 1.00i)13-s + (0.203 + 3.22i)14-s + (−4.33 − 2.91i)15-s + (−0.617 + 0.746i)16-s + (3.03 − 6.44i)17-s + ⋯
L(s)  = 1  + (0.282 − 0.445i)2-s + (0.982 − 0.922i)3-s + (0.307 + 0.652i)4-s + (−0.223 − 0.974i)5-s + (−0.133 − 0.698i)6-s + (−1.32 + 0.963i)7-s + (0.901 + 0.113i)8-s + (0.0511 − 0.813i)9-s + (−0.497 − 0.176i)10-s + (−0.763 + 1.20i)11-s + (0.903 + 0.357i)12-s + (0.0176 − 0.280i)13-s + (0.0543 + 0.863i)14-s + (−1.11 − 0.751i)15-s + (−0.154 + 0.186i)16-s + (0.735 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33424 - 0.682868i\)
\(L(\frac12)\) \(\approx\) \(1.33424 - 0.682868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.498 + 2.17i)T \)
good2 \( 1 + (-0.399 + 0.630i)T + (-0.851 - 1.80i)T^{2} \)
3 \( 1 + (-1.70 + 1.59i)T + (0.188 - 2.99i)T^{2} \)
7 \( 1 + (3.50 - 2.54i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.53 - 3.98i)T + (-4.68 - 9.95i)T^{2} \)
13 \( 1 + (-0.0635 + 1.00i)T + (-12.8 - 1.62i)T^{2} \)
17 \( 1 + (-3.03 + 6.44i)T + (-10.8 - 13.0i)T^{2} \)
19 \( 1 + (0.225 + 0.212i)T + (1.19 + 18.9i)T^{2} \)
23 \( 1 + (-2.47 - 0.635i)T + (20.1 + 11.0i)T^{2} \)
29 \( 1 + (2.93 + 1.61i)T + (15.5 + 24.4i)T^{2} \)
31 \( 1 + (-0.648 + 1.37i)T + (-19.7 - 23.8i)T^{2} \)
37 \( 1 + (3.81 - 4.61i)T + (-6.93 - 36.3i)T^{2} \)
41 \( 1 + (1.52 - 0.391i)T + (35.9 - 19.7i)T^{2} \)
43 \( 1 + (-2.70 + 8.31i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (3.85 - 0.486i)T + (45.5 - 11.6i)T^{2} \)
53 \( 1 + (-0.548 + 2.87i)T + (-49.2 - 19.5i)T^{2} \)
59 \( 1 + (-7.84 - 3.10i)T + (43.0 + 40.3i)T^{2} \)
61 \( 1 + (10.1 + 2.60i)T + (53.4 + 29.3i)T^{2} \)
67 \( 1 + (-3.30 + 1.81i)T + (35.9 - 56.5i)T^{2} \)
71 \( 1 + (-7.73 + 0.977i)T + (68.7 - 17.6i)T^{2} \)
73 \( 1 + (-5.38 + 2.13i)T + (53.2 - 49.9i)T^{2} \)
79 \( 1 + (3.01 - 2.82i)T + (4.96 - 78.8i)T^{2} \)
83 \( 1 + (10.6 + 10.0i)T + (5.21 + 82.8i)T^{2} \)
89 \( 1 + (3.89 - 1.54i)T + (64.8 - 60.9i)T^{2} \)
97 \( 1 + (-7.39 - 4.06i)T + (51.9 + 81.8i)T^{2} \)
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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

−12.97821691794801567453908384486, −12.50870547604556347138825406729, −11.81631107312970299885952927183, −9.861016022836821898831184887247, −8.900410551958504543102975358585, −7.81741420976773415117755532106, −7.03594121820158998767398713227, −5.11944372825759485168282539181, −3.25114396783998462020993118400, −2.26019054334143362306572608427, 3.08732231855109843203098099340, 3.93811580906833868071833502028, 5.88267039687788830349061781975, 6.88546392445369521854366480925, 8.112730871525661401474531811285, 9.618378656329811942718623613962, 10.43185566320119322863922977741, 10.89269190107197984571610794060, 12.99750241532304249507848530651, 13.96297607712151389437190211934

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