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Properties

Label 2-5e3-25.21-c1-0-2
Degree $2$
Conductor $125$
Sign $0.939 + 0.343i$
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  + (0.0566 − 0.174i)2-s + (−1.19 − 0.865i)3-s + (1.59 + 1.15i)4-s + (−0.218 + 0.158i)6-s + 3.26·7-s + (0.587 − 0.427i)8-s + (−0.257 − 0.792i)9-s + (0.618 − 1.90i)11-s + (−0.894 − 2.75i)12-s + (0.0915 + 0.281i)13-s + (0.184 − 0.568i)14-s + (1.17 + 3.61i)16-s + (−4.17 + 3.03i)17-s − 0.152·18-s + (−1.39 + 1.01i)19-s + ⋯
L(s)  = 1  + (0.0400 − 0.123i)2-s + (−0.687 − 0.499i)3-s + (0.795 + 0.577i)4-s + (−0.0890 + 0.0646i)6-s + 1.23·7-s + (0.207 − 0.150i)8-s + (−0.0858 − 0.264i)9-s + (0.186 − 0.573i)11-s + (−0.258 − 0.794i)12-s + (0.0254 + 0.0781i)13-s + (0.0493 − 0.151i)14-s + (0.293 + 0.903i)16-s + (−1.01 + 0.736i)17-s − 0.0359·18-s + (−0.321 + 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09660 - 0.194124i\)
\(L(\frac12)\) \(\approx\) \(1.09660 - 0.194124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0566 + 0.174i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.19 + 0.865i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.0915 - 0.281i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.17 - 3.03i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.39 - 1.01i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.271 - 0.836i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.78 + 3.47i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.93 - 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.49 + 7.69i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.313 + 0.965i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.24T + 43T^{2} \)
47 \( 1 + (-3.41 - 2.48i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.55 + 4.76i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.83 - 5.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.282 + 0.870i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.57 + 4.04i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.82 - 3.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.72 - 8.40i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.27 - 4.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.7 + 8.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.32 + 7.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.44 + 3.95i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.01249949673815935351745768090, −12.19345271266321141975745996878, −11.21561870721458646655522883685, −10.90769616154320059731686557905, −8.884861431076541705970464253489, −7.81925444396419464611860577418, −6.73172306751603999865226921035, −5.66323958168557440638883451188, −3.89264974497479207267070931410, −1.85881092676892670588487740889, 2.05853639616035965681237147004, 4.61678595940197664571459105218, 5.41240168674276780142563663111, 6.76848024863612655410634158421, 7.921405689239858705021277503258, 9.448913642242648047327529248016, 10.78241026821094122367765405391, 11.12106642460005228666197947207, 12.04895845413495995031010731115, 13.62550957236470618439509946807

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