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Properties

Label 1-4235-4235.984-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.640 + 0.768i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (0.683 − 0.730i)2-s + (−0.669 + 0.743i)3-s + (−0.0665 − 0.997i)4-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (−0.104 − 0.994i)9-s + (0.786 + 0.618i)12-s + (0.466 + 0.884i)13-s + (−0.991 + 0.132i)16-s + (0.948 − 0.318i)17-s + (−0.797 − 0.603i)18-s + (−0.595 + 0.803i)19-s + (−0.723 + 0.690i)23-s + (0.988 − 0.151i)24-s + (0.964 + 0.263i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.683 − 0.730i)2-s + (−0.669 + 0.743i)3-s + (−0.0665 − 0.997i)4-s + (0.0855 + 0.996i)6-s + (−0.774 − 0.633i)8-s + (−0.104 − 0.994i)9-s + (0.786 + 0.618i)12-s + (0.466 + 0.884i)13-s + (−0.991 + 0.132i)16-s + (0.948 − 0.318i)17-s + (−0.797 − 0.603i)18-s + (−0.595 + 0.803i)19-s + (−0.723 + 0.690i)23-s + (0.988 − 0.151i)24-s + (0.964 + 0.263i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.640 + 0.768i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (984, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.640 + 0.768i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09502501748 + 0.2028884228i\)
\(L(\frac12)\) \(\approx\) \(0.09502501748 + 0.2028884228i\)
\(L(1)\) \(\approx\) \(0.9498937166 - 0.2072582744i\)
\(L(1)\) \(\approx\) \(0.9498937166 - 0.2072582744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.683 - 0.730i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.466 + 0.884i)T \)
17 \( 1 + (0.948 - 0.318i)T \)
19 \( 1 + (-0.595 + 0.803i)T \)
23 \( 1 + (-0.723 + 0.690i)T \)
29 \( 1 + (-0.736 - 0.676i)T \)
31 \( 1 + (0.272 - 0.962i)T \)
37 \( 1 + (-0.640 + 0.768i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.797 + 0.603i)T \)
53 \( 1 + (0.991 + 0.132i)T \)
59 \( 1 + (0.483 - 0.875i)T \)
61 \( 1 + (-0.290 + 0.956i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.997 - 0.0760i)T \)
79 \( 1 + (0.449 - 0.893i)T \)
83 \( 1 + (-0.941 - 0.336i)T \)
89 \( 1 + (-0.995 - 0.0950i)T \)
97 \( 1 + (0.362 - 0.931i)T \)
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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

−17.98860941897635897305254767882, −17.449180713658504417336120616743, −16.61835116636166164683315808011, −16.278666086527043534461159467226, −15.41456706611713655624488281603, −14.70596913521164803716687592247, −14.028084946330236562699475244, −13.27748886476624219344067916188, −12.75336029251806208441824644983, −12.25844245669600717947667033175, −11.49020768561214412348722388620, −10.757422803396921357033149440937, −10.03662927530343103142921944218, −8.652000631801830806289364719819, −8.33320372608545990986211590271, −7.44407089232098406813425203655, −6.89315874604722577615299467455, −6.15911070818873304510398472165, −5.53026803448043415756147275522, −4.99471111847322494155243775258, −4.03544682295370965720354056613, −3.18027680743766943530943479166, −2.34183374799543544909356034445, −1.29517063119997712343002172968, −0.05426286593782393457265366379, 1.21751537210985704958717842269, 1.97505691366259258202083343412, 3.11016949193951257148827243863, 3.84612508102380307623366823562, 4.28122665887704396200511936437, 5.15650021034362283711651820975, 5.90212287587035528439269714780, 6.27108995545071607444942337922, 7.31768684602824895380949881300, 8.43721868322589722262678363539, 9.33679974342991926488128119678, 9.903856889523861737616735363839, 10.38297990440626399512765610224, 11.335640905795788912567714881301, 11.676616427491588954974566042382, 12.264289861562606542202480086986, 13.12673369682984775266436283593, 13.86412834772075543601419556844, 14.55307702115738835371725417919, 15.12476217320956676799996335381, 15.89409914121889128705069237959, 16.48155272710527098900235907607, 17.17059686678236112879859771042, 18.05427084675050230273735412242, 18.77502511340083165100919265195

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