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Properties

Label 1-4235-4235.1272-r1-0-0
Degree $1$
Conductor $4235$
Sign $-0.897 - 0.440i$
Analytic cond. $455.113$
Root an. cond. $455.113$
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  + (−0.263 − 0.964i)2-s + (0.406 + 0.913i)3-s + (−0.861 + 0.508i)4-s + (0.774 − 0.633i)6-s + (0.717 + 0.696i)8-s + (−0.669 + 0.743i)9-s + (−0.814 − 0.580i)12-s + (0.676 + 0.736i)13-s + (0.483 − 0.875i)16-s + (0.524 − 0.851i)17-s + (0.893 + 0.449i)18-s + (−0.380 + 0.924i)19-s + (0.189 − 0.981i)23-s + (−0.345 + 0.938i)24-s + (0.532 − 0.846i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (−0.263 − 0.964i)2-s + (0.406 + 0.913i)3-s + (−0.861 + 0.508i)4-s + (0.774 − 0.633i)6-s + (0.717 + 0.696i)8-s + (−0.669 + 0.743i)9-s + (−0.814 − 0.580i)12-s + (0.676 + 0.736i)13-s + (0.483 − 0.875i)16-s + (0.524 − 0.851i)17-s + (0.893 + 0.449i)18-s + (−0.380 + 0.924i)19-s + (0.189 − 0.981i)23-s + (−0.345 + 0.938i)24-s + (0.532 − 0.846i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.897 - 0.440i$
Analytic conductor: \(455.113\)
Root analytic conductor: \(455.113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (1:\ ),\ -0.897 - 0.440i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1281397778 - 0.5519315025i\)
\(L(\frac12)\) \(\approx\) \(0.1281397778 - 0.5519315025i\)
\(L(1)\) \(\approx\) \(0.9160830039 - 0.1145513412i\)
\(L(1)\) \(\approx\) \(0.9160830039 - 0.1145513412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.263 - 0.964i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
13 \( 1 + (0.676 + 0.736i)T \)
17 \( 1 + (0.524 - 0.851i)T \)
19 \( 1 + (-0.380 + 0.924i)T \)
23 \( 1 + (0.189 - 0.981i)T \)
29 \( 1 + (0.941 - 0.336i)T \)
31 \( 1 + (0.595 - 0.803i)T \)
37 \( 1 + (0.662 + 0.749i)T \)
41 \( 1 + (-0.362 - 0.931i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.893 - 0.449i)T \)
53 \( 1 + (0.875 - 0.483i)T \)
59 \( 1 + (-0.625 - 0.780i)T \)
61 \( 1 + (-0.710 + 0.703i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.572 - 0.820i)T \)
79 \( 1 + (-0.830 - 0.556i)T \)
83 \( 1 + (0.389 + 0.921i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (0.170 + 0.985i)T \)
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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

−18.34297333297876531172525122639, −17.765374907130837988588677530865, −17.321043740248653257275492294076, −16.55231094602696952233258011371, −15.61158291756119882438500923941, −15.18559873075897663264898109332, −14.47318613780912330720469893320, −13.75502538680950961426126787390, −13.21311156145367692308730600505, −12.67867398278982751478113298279, −11.805540851294808394858478457071, −10.8127593472834656363901516818, −10.149090217491935860524716949411, −9.17104869630876324450495493418, −8.63371887474030675380671064711, −8.021232870229530057100971813752, −7.42574232324988293490892554937, −6.65360191915316478855519565203, −6.065365129548197840370390567930, −5.41996155589170763574866420731, −4.444828441607952331130742921749, −3.48417276451636874620077796806, −2.74417104949543082921673322755, −1.42756335146948323102162866380, −1.00910573352629862864782578724, 0.09645399185581201361523561521, 1.13472933721014936205324713219, 2.15949212422508113194700834329, 2.80196323963329043129404639993, 3.620878742837267978730373356617, 4.24426241240701179532891027141, 4.85087849005678649474734227087, 5.739125203908711421343808515782, 6.75247399817368474094592720783, 7.95368666991475853161379234108, 8.35464952446451243767682377278, 9.1237468591324456970806367464, 9.69387955398020999630805790948, 10.408481547349283249152070531051, 10.82661647727356555671163966341, 11.904486066683647178504980761762, 12.00996019100737071780963853894, 13.35671121187762105165221778807, 13.71500057495489368511923527424, 14.42875523659052209684963732524, 15.105087683758887766013388581542, 16.07020487223291289128446168648, 16.65016021345502149616111862674, 17.087997348366104290206179683789, 18.15896222951397492227573628781

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