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Properties

Label 1-4235-4235.639-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.855 - 0.517i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2595358347 - 0.9310649580i\)
\(L(\frac12)\) \(\approx\) \(0.2595358347 - 0.9310649580i\)
\(L(1)\) \(\approx\) \(0.6647649795 - 0.4335658919i\)
\(L(1)\) \(\approx\) \(0.6647649795 - 0.4335658919i\)

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

−18.72605068090760986814447829942, −17.79552658471969885898355664636, −17.1773471065143473002681572957, −16.57800195048315334597110158287, −16.01110157337838467594950355272, −15.298363109799550659014129008883, −14.63816843568669899948612355664, −14.42041057230924812139688477459, −13.39407585127692600447719839539, −12.66435557434607504608971779867, −11.489807095354664052221112214939, −10.83316353849035363749476751151, −10.27624119563801066141722279806, −9.5157075766396265944801519715, −9.05992088363707834146233720146, −8.38408665427898210962565435549, −7.494737066055396222833386656757, −7.1455604590486190555360395675, −5.98270718313334349717562099560, −5.17699193771367895712671498403, −4.83591895435008934346105433118, −3.74746935116260436922081800131, −2.79803029332802885152165717025, −2.11066822063351895839647950787, −0.86452033156867393834133026283, 0.3813839923810622990639944364, 1.498019319594685561753745428294, 1.99288211784569190264541080877, 2.832236990688668152336971739597, 3.56221509895997344015860518540, 4.33264173981224070669314217921, 5.49510720446426904727779104630, 6.50782329865926732581590917063, 7.1587575413011923203707461504, 7.72896728712631734177126150094, 8.56318051406369618352899248245, 8.913352982723372495063610334, 9.92425991133503142357802206010, 10.33474318101468836433658582002, 11.44180072613242131358681140668, 11.928039400131393610909195762157, 12.58282885671758714538986854801, 13.12101128467170637264710346274, 13.8476007875536437496119462230, 14.73289499240485858595959242371, 15.17800191873431356424555303284, 16.47611427682150945938402491234, 16.959430359446127465774030928016, 17.45747991013816706584552335666, 18.330034099637961470199623135005

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