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Properties

Label 1-4235-4235.1504-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.983 - 0.183i$
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  + (−0.921 + 0.389i)2-s + (−0.809 + 0.587i)3-s + (0.696 − 0.717i)4-s + (0.516 − 0.856i)6-s + (−0.362 + 0.931i)8-s + (0.309 − 0.951i)9-s + (−0.142 + 0.989i)12-s + (−0.897 − 0.441i)13-s + (−0.0285 − 0.999i)16-s + (0.736 − 0.676i)17-s + (0.0855 + 0.996i)18-s + (0.198 + 0.980i)19-s + (0.959 + 0.281i)23-s + (−0.254 − 0.967i)24-s + (0.998 + 0.0570i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.921 + 0.389i)2-s + (−0.809 + 0.587i)3-s + (0.696 − 0.717i)4-s + (0.516 − 0.856i)6-s + (−0.362 + 0.931i)8-s + (0.309 − 0.951i)9-s + (−0.142 + 0.989i)12-s + (−0.897 − 0.441i)13-s + (−0.0285 − 0.999i)16-s + (0.736 − 0.676i)17-s + (0.0855 + 0.996i)18-s + (0.198 + 0.980i)19-s + (0.959 + 0.281i)23-s + (−0.254 − 0.967i)24-s + (0.998 + 0.0570i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.983 - 0.183i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1504, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.983 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7951544336 - 0.07349818666i\)
\(L(\frac12)\) \(\approx\) \(0.7951544336 - 0.07349818666i\)
\(L(1)\) \(\approx\) \(0.5855218578 + 0.1093617728i\)
\(L(1)\) \(\approx\) \(0.5855218578 + 0.1093617728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.921 + 0.389i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.897 - 0.441i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.870 - 0.491i)T \)
31 \( 1 + (0.985 + 0.170i)T \)
37 \( 1 + (0.466 - 0.884i)T \)
41 \( 1 + (0.974 + 0.226i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.0855 - 0.996i)T \)
53 \( 1 + (0.0285 - 0.999i)T \)
59 \( 1 + (-0.974 + 0.226i)T \)
61 \( 1 + (-0.921 - 0.389i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (-0.610 - 0.791i)T \)
79 \( 1 + (-0.774 + 0.633i)T \)
83 \( 1 + (0.564 + 0.825i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.254 - 0.967i)T \)
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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

−18.415653615308499510262597069766, −17.615294027659231741502598925325, −17.19335333131019478194960803676, −16.714216968419586067118033687353, −15.93283223386804774197468936887, −15.2554783155756866451703186675, −14.26902986516123024089915604952, −13.41630019466269058007286802650, −12.48499134265986998170189821542, −12.34411501777635856283300839818, −11.42596333882613192950154229093, −10.91698673166216807356780959864, −10.23015136938619003011036295353, −9.50819134192893514022037636913, −8.765639084720666383953668093008, −7.81737632336692075404502728675, −7.41814998994947296957724756316, −6.57758626561583615702491417616, −6.0874725690706926128985932623, −4.92869541863580329808206649954, −4.33644157959542408289241350677, −2.94668845887285481715959607085, −2.49783280412807669466893153824, −1.36341142802985805893262580386, −0.8392388934021368704803510075, 0.5039112171090392713137743905, 1.208903160479649183501056318, 2.458147948905726344691944630242, 3.25146768646322005266065054291, 4.39721345257544603690812519426, 5.21564665603646659948867293749, 5.6908169100717067365177096015, 6.49075428253943465985464360549, 7.27670111892067619492148219656, 7.848314580522732767167726941118, 8.78046950469781359580501851009, 9.64635655691184889465396579294, 9.93144508805336203687156306906, 10.65954189992251970731349456054, 11.344658448302746832942433150553, 12.08717967872675044329113266321, 12.52997051074245334665721220798, 13.86081982960506018899838579758, 14.58184923081564048828417030776, 15.1651298100443796221139101788, 15.89159036604612119105392856108, 16.36395582013715224044577929528, 17.12618188462025895762109155164, 17.45746495800160821277264569787, 18.232847812981667043405788396462

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