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Properties

Label 1-4025-4025.37-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.895 - 0.444i$
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  + (0.986 + 0.161i)2-s + (0.0380 − 0.999i)3-s + (0.948 + 0.318i)4-s + (0.198 − 0.980i)6-s + (0.884 + 0.466i)8-s + (−0.997 − 0.0760i)9-s + (0.710 + 0.703i)11-s + (0.353 − 0.935i)12-s + (0.856 − 0.516i)13-s + (0.797 + 0.603i)16-s + (0.662 − 0.749i)17-s + (−0.971 − 0.235i)18-s + (0.861 + 0.508i)19-s + (0.587 + 0.809i)22-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.986 + 0.161i)2-s + (0.0380 − 0.999i)3-s + (0.948 + 0.318i)4-s + (0.198 − 0.980i)6-s + (0.884 + 0.466i)8-s + (−0.997 − 0.0760i)9-s + (0.710 + 0.703i)11-s + (0.353 − 0.935i)12-s + (0.856 − 0.516i)13-s + (0.797 + 0.603i)16-s + (0.662 − 0.749i)17-s + (−0.971 − 0.235i)18-s + (0.861 + 0.508i)19-s + (0.587 + 0.809i)22-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.145665669 - 0.9717757533i\)
\(L(\frac12)\) \(\approx\) \(4.145665669 - 0.9717757533i\)
\(L(1)\) \(\approx\) \(2.220100239 - 0.3625829157i\)
\(L(1)\) \(\approx\) \(2.220100239 - 0.3625829157i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.986 + 0.161i)T \)
3 \( 1 + (0.0380 - 0.999i)T \)
11 \( 1 + (0.710 + 0.703i)T \)
13 \( 1 + (0.856 - 0.516i)T \)
17 \( 1 + (0.662 - 0.749i)T \)
19 \( 1 + (0.861 + 0.508i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (-0.532 - 0.846i)T \)
37 \( 1 + (0.0760 - 0.997i)T \)
41 \( 1 + (-0.0285 + 0.999i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (-0.406 + 0.913i)T \)
53 \( 1 + (-0.730 - 0.683i)T \)
59 \( 1 + (0.999 - 0.0190i)T \)
61 \( 1 + (0.830 + 0.556i)T \)
67 \( 1 + (-0.836 - 0.548i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.803 + 0.595i)T \)
79 \( 1 + (-0.905 + 0.424i)T \)
83 \( 1 + (-0.791 + 0.610i)T \)
89 \( 1 + (0.640 + 0.768i)T \)
97 \( 1 + (-0.791 - 0.610i)T \)
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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

−18.79731385916323711379042588477, −17.64563736741527260495334466692, −16.825330136011005755246034713040, −16.288251967892637804440946320582, −15.787141388104138153528276757250, −15.05165789082697039904694596100, −14.39122862155897044729140371650, −13.8430223647826456601397413789, −13.31115928332292700679734662103, −12.16225187754923741072331758634, −11.65845617897311918316184724646, −11.08357177268013312393184491580, −10.36463661454258217402678133107, −9.73752010441534520526124194340, −8.7903384107900748591481032177, −8.228699785241293087580528771623, −7.05393988963095325060395951157, −6.27515472761140122904201758311, −5.713353381437885102423173510588, −4.9335351232502340233740557933, −4.22857604729413570827906602490, −3.37745915976191935425325254207, −3.2114130983563858565844355890, −1.90629718932756875262036563071, −1.00160699170999348637454296995, 1.051289741188576203818468987509, 1.63378524940394987587564966151, 2.65235044477315981194742337390, 3.308471538800464587483804789, 4.07326970086945250854566841664, 5.15831868730005880866476643894, 5.6870149622530029896703158107, 6.48740175794726908565963277947, 7.05429979824403310345024311721, 7.7647857180487381003524033607, 8.33022648700345110345004673188, 9.39721415360846516454279956295, 10.25124666132315226754627320245, 11.377907772184090273865155498076, 11.61173423848335339882688241273, 12.46540322532154790344746041347, 12.92098703206791616036994751064, 13.62349813760787367465237430632, 14.388243345007243878290547466692, 14.613184509345781646202360100873, 15.67442189058531116091816299675, 16.32389022165495338459833911455, 16.96281260759351026992972278377, 17.882155692888754108556148663806, 18.24913651118836281272266228561

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