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Properties

Label 1-4235-4235.54-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.966 - 0.257i$
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  + (−0.888 − 0.458i)2-s + (−0.5 + 0.866i)3-s + (0.580 + 0.814i)4-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.995 + 0.0950i)12-s + (−0.415 + 0.909i)13-s + (−0.327 + 0.945i)16-s + (−0.723 + 0.690i)17-s + (0.0475 + 0.998i)18-s + (0.723 + 0.690i)19-s + (0.327 − 0.945i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + 27-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (−0.5 + 0.866i)3-s + (0.580 + 0.814i)4-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.995 + 0.0950i)12-s + (−0.415 + 0.909i)13-s + (−0.327 + 0.945i)16-s + (−0.723 + 0.690i)17-s + (0.0475 + 0.998i)18-s + (0.723 + 0.690i)19-s + (0.327 − 0.945i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.966 - 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7350525002 - 0.09643357334i\)
\(L(\frac12)\) \(\approx\) \(0.7350525002 - 0.09643357334i\)
\(L(1)\) \(\approx\) \(0.5965255344 + 0.05332115508i\)
\(L(1)\) \(\approx\) \(0.5965255344 + 0.05332115508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.888 - 0.458i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (-0.580 + 0.814i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.0475 - 0.998i)T \)
53 \( 1 + (0.327 + 0.945i)T \)
59 \( 1 + (0.888 - 0.458i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.981 + 0.189i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.235 - 0.971i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−17.9805818286979945576080729798, −17.71916061768352817178982362652, −17.51554476371092443121075304389, −16.27770218123731870088192474520, −16.03348073985924032355177136199, −15.16848853554127515271069951107, −14.39711614038037356039451487757, −13.59861082011742709009268985114, −13.052905870906339690760946636, −12.03659967851649524446399269060, −11.51705825125843541268955729830, −10.87947876824136445727794721115, −10.12333311557866661638102604608, −9.37058984251018471387883266678, −8.592167895456320669089831026075, −7.85665589268820636379394928972, −7.26041122386433586332752354677, −6.75941728417541700161945780947, −5.89442841688626201877038413227, −5.28444031528836908119959726301, −4.601904149136750143817675653362, −2.88599773314670797707042677601, −2.52501427203114623587529448520, −1.304078122115836133872271548625, −0.7478496248406435729725003329, 0.441789026257258074035975894182, 1.5306485838939059921905741828, 2.47464564196261468038153874590, 3.28530660540104069540857084994, 4.16168495128688570922954029447, 4.665210218603817380319292440683, 5.79887377719620991736759577874, 6.57762091548258016644632794008, 7.12929239905549319052070557231, 8.32873698000886560475768831656, 8.72169547123733859073118157030, 9.52040203722378242307182939245, 10.192609793951718082919224591550, 10.582580934676374949512441067902, 11.44713502511589781314887732142, 11.99700877870388213129296986808, 12.47578063908859648804194358499, 13.586104989637467243165178288131, 14.38806507353279485832022480080, 15.255948765358419776973469414633, 15.84235075374692031058788731499, 16.42481681623241324128592121809, 17.183543414638571536675282067680, 17.41568664138808734389170635784, 18.36551523431229170907801933245

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