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Properties

Label 1-4025-4025.4-r0-0-0
Degree $1$
Conductor $4025$
Sign $-0.736 - 0.676i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.290 − 0.956i)2-s + (−0.161 + 0.986i)3-s + (−0.830 − 0.556i)4-s + (0.897 + 0.441i)6-s + (−0.774 + 0.633i)8-s + (−0.948 − 0.318i)9-s + (0.820 − 0.572i)11-s + (0.683 − 0.730i)12-s + (−0.941 + 0.336i)13-s + (0.380 + 0.924i)16-s + (0.0665 + 0.997i)17-s + (−0.580 + 0.814i)18-s + (0.640 − 0.768i)19-s + (−0.309 − 0.951i)22-s + (−0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.290 − 0.956i)2-s + (−0.161 + 0.986i)3-s + (−0.830 − 0.556i)4-s + (0.897 + 0.441i)6-s + (−0.774 + 0.633i)8-s + (−0.948 − 0.318i)9-s + (0.820 − 0.572i)11-s + (0.683 − 0.730i)12-s + (−0.941 + 0.336i)13-s + (0.380 + 0.924i)16-s + (0.0665 + 0.997i)17-s + (−0.580 + 0.814i)18-s + (0.640 − 0.768i)19-s + (−0.309 − 0.951i)22-s + (−0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ -0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2811866370 - 0.7210986019i\)
\(L(\frac12)\) \(\approx\) \(0.2811866370 - 0.7210986019i\)
\(L(1)\) \(\approx\) \(0.8548085595 - 0.2525570481i\)
\(L(1)\) \(\approx\) \(0.8548085595 - 0.2525570481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.290 - 0.956i)T \)
3 \( 1 + (-0.161 + 0.986i)T \)
11 \( 1 + (0.820 - 0.572i)T \)
13 \( 1 + (-0.941 + 0.336i)T \)
17 \( 1 + (0.0665 + 0.997i)T \)
19 \( 1 + (0.640 - 0.768i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
31 \( 1 + (-0.935 - 0.353i)T \)
37 \( 1 + (0.948 + 0.318i)T \)
41 \( 1 + (-0.870 - 0.491i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.432 - 0.901i)T \)
59 \( 1 + (-0.761 - 0.647i)T \)
61 \( 1 + (0.988 + 0.151i)T \)
67 \( 1 + (-0.483 + 0.875i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (-0.999 + 0.0380i)T \)
79 \( 1 + (0.879 + 0.475i)T \)
83 \( 1 + (0.736 + 0.676i)T \)
89 \( 1 + (0.548 - 0.836i)T \)
97 \( 1 + (0.736 - 0.676i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.4912878796994969050418807093, −17.9949576073072439022439977636, −17.32046074802930824944133255112, −16.69505367269389718007491187965, −16.26473561841874514770310043057, −15.01048965005216375838493314760, −14.75923728241488754453893872900, −13.950397710697839672412272726269, −13.41713277085740335650468971988, −12.61922075322921380134206315052, −12.03198456191632022083230830005, −11.62486995462910120072799537556, −10.34513222818436529998091559015, −9.4177994149795271257137884013, −8.96768007108322394090840414766, −7.87932453709480240918625237445, −7.41838978377322891834457082496, −6.97324804650501778332367192497, −6.10611885496651193313758758188, −5.43223175114884312779958974826, −4.81421932578904861530734186350, −3.79203675217931900820725050962, −2.95875598707010003018341493077, −1.99249215852542163052535743420, −0.94187002394666355111709823572, 0.23528002190359758960965240595, 1.4298687124360388676425709232, 2.38000941445026795410468826092, 3.26561281428949407852155101511, 3.84135693612701433658451589465, 4.50966706370790178720615862228, 5.29345813531747684155663910583, 5.877111092630386704692312799158, 6.79501608998056416804558616276, 7.98631645751308923646566180339, 8.94004338528271143317040602034, 9.278038532605607954573568651, 10.01388397137093866441284690092, 10.646544690354902771106163587057, 11.54248321902443891909212758085, 11.62867295105378234946377153561, 12.65365135691515842668360272029, 13.38946707940117545650365672045, 14.20560189980319795757805240541, 14.79468784240527340315923914028, 15.17707853321885550459298818812, 16.29320861290625392809934072735, 16.92599603317968644880067026749, 17.435710343501414618785820814093, 18.301843092417397899231569525279

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