Properties

Label 1-4235-4235.1684-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.295 + 0.955i$
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  + (0.995 + 0.0950i)2-s + (0.5 + 0.866i)3-s + (0.981 + 0.189i)4-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.327 + 0.945i)12-s + (0.654 − 0.755i)13-s + (0.928 + 0.371i)16-s + (0.888 + 0.458i)17-s + (−0.580 + 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.928 − 0.371i)23-s + (0.235 + 0.971i)24-s + (0.723 − 0.690i)26-s − 27-s + ⋯
L(s)  = 1  + (0.995 + 0.0950i)2-s + (0.5 + 0.866i)3-s + (0.981 + 0.189i)4-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (0.327 + 0.945i)12-s + (0.654 − 0.755i)13-s + (0.928 + 0.371i)16-s + (0.888 + 0.458i)17-s + (−0.580 + 0.814i)18-s + (−0.888 + 0.458i)19-s + (−0.928 − 0.371i)23-s + (0.235 + 0.971i)24-s + (0.723 − 0.690i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.295 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.557139613 + 3.468122909i\)
\(L(\frac12)\) \(\approx\) \(2.557139613 + 3.468122909i\)
\(L(1)\) \(\approx\) \(2.088554797 + 1.069981661i\)
\(L(1)\) \(\approx\) \(2.088554797 + 1.069981661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.995 + 0.0950i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.654 - 0.755i)T \)
17 \( 1 + (0.888 + 0.458i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (-0.981 + 0.189i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.580 - 0.814i)T \)
53 \( 1 + (-0.928 + 0.371i)T \)
59 \( 1 + (-0.995 + 0.0950i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.786 - 0.618i)T \)
79 \( 1 + (0.235 - 0.971i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−18.362385537346987150070427767498, −17.45289829985295850388738835963, −16.79589592370260495293432895895, −15.86190138291210610408411336993, −15.41197847849646183842870845012, −14.466737066967499330852949285306, −13.92681690514983635467268935006, −13.67216015659949563065067935654, −12.5471294139987995297177158447, −12.42197817041927134088013636937, −11.46111042400913305658668409251, −10.99871477297490400059428235457, −9.913656730682861201506322336047, −9.184893151170718358146734315355, −8.215326360685870515524272977451, −7.64481586597378556447990426140, −6.85451034747646420696273097339, −6.23898778410694561307816840582, −5.67192848745591387108337963387, −4.60862950004284583776841639214, −3.83713375035453225490616315859, −3.202508332998868089107296944423, −2.2404351114264380254711137462, −1.7837349653701857338283000219, −0.71626468279551446508851403131, 1.3418399019037790065768341058, 2.23324914995871076967279898437, 3.17608537365852895786926061620, 3.56787615791155544718689116191, 4.38557319511357388194706127069, 5.05516780597731110977113448318, 5.8470905141004905961511341670, 6.387511280434644337483738302999, 7.528383341861482880752105631432, 8.19334121585663969841311228261, 8.693432318196184806031582250595, 9.88749145239918874282124247824, 10.5387612671095459380210210508, 10.82404021758194081656280606970, 11.9224225019075803820992248960, 12.525307133062968741779786584769, 13.227875693569359409377948799602, 14.03389284975643013282299844789, 14.515303703290128574387923749758, 15.03545905900713657520142621819, 15.90750730573867969892447823819, 16.16391096294928385314341200980, 16.94718911273362872800207345852, 17.72360181343586910832706175362, 18.75132808400850734198587991481

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