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Properties

Label 1-4235-4235.1979-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.868 + 0.495i$
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  + (−0.786 + 0.618i)2-s + (−0.5 + 0.866i)3-s + (0.235 − 0.971i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 + 0.690i)12-s + (0.959 − 0.281i)13-s + (−0.888 − 0.458i)16-s + (−0.981 + 0.189i)17-s + (0.928 + 0.371i)18-s + (0.981 + 0.189i)19-s + (0.888 + 0.458i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s + 27-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (−0.5 + 0.866i)3-s + (0.235 − 0.971i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 + 0.690i)12-s + (0.959 − 0.281i)13-s + (−0.888 − 0.458i)16-s + (−0.981 + 0.189i)17-s + (0.928 + 0.371i)18-s + (0.981 + 0.189i)19-s + (0.888 + 0.458i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s + 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.868 + 0.495i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8985850532 + 0.2380866996i\)
\(L(\frac12)\) \(\approx\) \(0.8985850532 + 0.2380866996i\)
\(L(1)\) \(\approx\) \(0.6226901714 + 0.2583447130i\)
\(L(1)\) \(\approx\) \(0.6226901714 + 0.2583447130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.786 + 0.618i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.981 + 0.189i)T \)
19 \( 1 + (0.981 + 0.189i)T \)
23 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.723 + 0.690i)T \)
37 \( 1 + (-0.235 - 0.971i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.928 - 0.371i)T \)
53 \( 1 + (0.888 - 0.458i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (0.928 - 0.371i)T \)
67 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.0475 - 0.998i)T \)
79 \( 1 + (0.995 - 0.0950i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (0.327 - 0.945i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−18.336030859058801440771883045031, −17.8306819711697144107798911592, −17.19181667367490291180778500298, −16.42854856371727865147261834313, −15.994618305714194409891038501302, −15.0462393171598510245881987142, −13.82589466212817078597682370987, −13.42975310886340380950384845622, −12.76393585308101058800940363727, −12.02983176196865179643646100301, −11.379167453287257601536537706060, −10.95681204721157583164895244164, −10.241499842157942890315763039848, −9.20449681213841480974527364379, −8.68304164760542968801866072172, −8.01023300761423810014220535344, −7.03906413168137581753867853352, −6.81400364537297223198397761326, −5.8228997923112287980316545648, −4.890227229017768625397639689205, −3.97714350455307019134812193109, −2.969698893561175825090530278039, −2.33539275756207385338345804185, −1.36181535240891678200430667747, −0.82067613233917777941846808459, 0.51992313278867627668497396690, 1.40011396094869004426996936312, 2.57523821763748952158322256033, 3.59632748204669388128150790725, 4.37560524432146968936316078946, 5.35383778431734115681798820420, 5.69168654183971892434791470600, 6.57946941383461750594641158693, 7.20654361983751540356730404680, 8.163134893210614901058812718397, 9.00242917712072719034216383852, 9.21430687095022996706263482410, 10.28281070911928348719331122953, 10.66347344151883486228152622480, 11.347355251538868090965261098919, 11.9673148282503159904546564617, 13.135389471630927823968097467025, 13.86450804611615143310969569822, 14.66700156310121934428979284238, 15.29842840445923786322921316925, 15.98764302166020597885528403739, 16.16353646711018087458528123029, 17.1304540390234862852092168581, 17.75119058042597131325524360436, 18.04347121691699947904574150714

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