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Properties

Label 1-4235-4235.1139-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.716 + 0.697i$
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  + (−0.991 − 0.132i)2-s + (−0.978 − 0.207i)3-s + (0.964 + 0.263i)4-s + (0.941 + 0.336i)6-s + (−0.921 − 0.389i)8-s + (0.913 + 0.406i)9-s + (−0.888 − 0.458i)12-s + (0.362 − 0.931i)13-s + (0.861 + 0.508i)16-s + (−0.272 − 0.962i)17-s + (−0.851 − 0.524i)18-s + (−0.830 − 0.556i)19-s + (0.995 − 0.0950i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (−0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (−0.991 − 0.132i)2-s + (−0.978 − 0.207i)3-s + (0.964 + 0.263i)4-s + (0.941 + 0.336i)6-s + (−0.921 − 0.389i)8-s + (0.913 + 0.406i)9-s + (−0.888 − 0.458i)12-s + (0.362 − 0.931i)13-s + (0.861 + 0.508i)16-s + (−0.272 − 0.962i)17-s + (−0.851 − 0.524i)18-s + (−0.830 − 0.556i)19-s + (0.995 − 0.0950i)23-s + (0.820 + 0.572i)24-s + (−0.483 + 0.875i)26-s + (−0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4692727766 + 0.1908305546i\)
\(L(\frac12)\) \(\approx\) \(0.4692727766 + 0.1908305546i\)
\(L(1)\) \(\approx\) \(0.5011441903 - 0.05692172656i\)
\(L(1)\) \(\approx\) \(0.5011441903 - 0.05692172656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.991 - 0.132i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.362 - 0.931i)T \)
17 \( 1 + (-0.272 - 0.962i)T \)
19 \( 1 + (-0.830 - 0.556i)T \)
23 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.449 + 0.893i)T \)
37 \( 1 + (0.935 + 0.353i)T \)
41 \( 1 + (-0.564 - 0.825i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.851 + 0.524i)T \)
53 \( 1 + (-0.861 + 0.508i)T \)
59 \( 1 + (0.432 + 0.901i)T \)
61 \( 1 + (0.380 - 0.924i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.466 + 0.884i)T \)
73 \( 1 + (-0.953 + 0.299i)T \)
79 \( 1 + (0.290 + 0.956i)T \)
83 \( 1 + (-0.198 + 0.980i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (0.0855 - 0.996i)T \)
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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

−18.18287420529628291277892067900, −17.56704437234612884872116847697, −16.85251227923605739070238260317, −16.57282243658275510253658001358, −15.84242406663300081858669776605, −15.022068306806282439991588545488, −14.654720849350504954305212818020, −13.32850977333477071859184550304, −12.69775274557492498809525285854, −11.80397555533408672142344556533, −11.3688464017991273792991019275, −10.66947603854084300726651730496, −10.12715999769535749940638188529, −9.37737758060028801922258078523, −8.66755020264729461425575085389, −7.941608204956595799828238597082, −7.01315811356216233845049771607, −6.380715363267810687118543361382, −6.006142399649537736357863851650, −4.967103452303256268988717968235, −4.176906381654919680788511453968, −3.26961588532509375809302597318, −1.97668610234664653571680995898, −1.474927300009012092364292170930, −0.31554101560476273565695622428, 0.77373325299870762492302384858, 1.396219999580752011552860736347, 2.538154107470869854030372244181, 3.17876421339081805280595368626, 4.43228824541092974403986358740, 5.19101723747126652091923174982, 6.00238613666164313524842174452, 6.80538667555208904866237377590, 7.14501084934178371843910222417, 8.15015153929431776904976318813, 8.71670640242273250110681984717, 9.66117354162290799237910244483, 10.2525712950610562737880600369, 11.05802603045447838845907438916, 11.264023688568337153941382979682, 12.221541423669818118652717270985, 12.80336990011130571013677297347, 13.40467675612314878888546302966, 14.60648193781327000545785322388, 15.45104647747153846857669326339, 15.914643170591481602313961071745, 16.58104320081145786418146847862, 17.263252686894195793780073941349, 17.82853668775182586577482604042, 18.24091805632049331590899177946

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