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Properties

Label 1-4235-4235.2749-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.169 - 0.985i$
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  + (0.723 + 0.690i)2-s + (−0.5 + 0.866i)3-s + (0.0475 + 0.998i)4-s + (−0.959 + 0.281i)6-s + (−0.654 + 0.755i)8-s + (−0.5 − 0.866i)9-s + (−0.888 − 0.458i)12-s + (−0.841 + 0.540i)13-s + (−0.995 + 0.0950i)16-s + (0.786 + 0.618i)17-s + (0.235 − 0.971i)18-s + (−0.786 + 0.618i)19-s + (0.995 − 0.0950i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s + 27-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (−0.5 + 0.866i)3-s + (0.0475 + 0.998i)4-s + (−0.959 + 0.281i)6-s + (−0.654 + 0.755i)8-s + (−0.5 − 0.866i)9-s + (−0.888 − 0.458i)12-s + (−0.841 + 0.540i)13-s + (−0.995 + 0.0950i)16-s + (0.786 + 0.618i)17-s + (0.235 − 0.971i)18-s + (−0.786 + 0.618i)19-s + (0.995 − 0.0950i)23-s + (−0.327 − 0.945i)24-s + (−0.981 − 0.189i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.169 - 0.985i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7436724078 + 0.8822264688i\)
\(L(\frac12)\) \(\approx\) \(-0.7436724078 + 0.8822264688i\)
\(L(1)\) \(\approx\) \(0.6518705182 + 0.8901762180i\)
\(L(1)\) \(\approx\) \(0.6518705182 + 0.8901762180i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (0.786 + 0.618i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (-0.0475 + 0.998i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (0.235 + 0.971i)T \)
53 \( 1 + (0.995 + 0.0950i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.580 - 0.814i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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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

−18.05681218335819923439323995242, −17.20072724984783369241737474643, −16.783048575316250579386594230687, −15.57644613004458906821158336686, −15.1487515526061590138512410522, −14.16482717449415247811259103554, −13.75272145054571180543908554351, −12.91333585362915470256313893108, −12.51623735571684759870739051914, −11.7907580641864760020166102242, −11.31300671683007648538933540566, −10.438755739472296712597800738690, −9.9543916961708671274945892323, −8.941264766235465086039646203784, −8.09167262377879977047852815681, −7.08398568945054968779403998692, −6.74178655637356389508394404053, −5.70226623652078555739115600555, −5.20162424599791070808397009835, −4.57615997764250074778835615840, −3.44969095593388089425117508555, −2.629386294379215852522540317756, −2.097224631276757342791358355339, −1.02784355282528999771945612284, −0.28559214012228619331104063180, 1.418074081861232160966737809930, 2.754897627767108600545000147868, 3.32627875058249067721468250190, 4.28691681397053631115230775582, 4.69837269827615342898797008623, 5.45103809051941820683895160629, 6.19166930129298155423562428440, 6.75718856596979017961720285322, 7.64181169020976923208248914272, 8.49484880314660371481891007295, 9.09379835102615271171928795005, 10.0409988720966592052324414808, 10.58839372033501930417511855498, 11.60385917888148593719081775565, 12.02882695856793749715289749737, 12.71080656357131465036006518266, 13.51177734606882334520333661777, 14.451117010364258810358800915308, 14.88187961521369581916580967177, 15.27821500308166193296508346851, 16.307246833353897923099925349542, 16.73302573757794481226851395699, 17.1051676928762887412681143662, 17.87157008551601387941112882966, 18.78355723379704429693754416507

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