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Properties

Label 1-4235-4235.129-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.245 - 0.969i$
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  + (0.640 − 0.768i)2-s + (0.913 + 0.406i)3-s + (−0.179 − 0.983i)4-s + (0.897 − 0.441i)6-s + (−0.870 − 0.491i)8-s + (0.669 + 0.743i)9-s + (0.235 − 0.971i)12-s + (0.0285 − 0.999i)13-s + (−0.935 + 0.353i)16-s + (0.999 + 0.0190i)17-s + (0.999 − 0.0380i)18-s + (0.820 + 0.572i)19-s + (−0.0475 − 0.998i)23-s + (−0.595 − 0.803i)24-s + (−0.749 − 0.662i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (0.640 − 0.768i)2-s + (0.913 + 0.406i)3-s + (−0.179 − 0.983i)4-s + (0.897 − 0.441i)6-s + (−0.870 − 0.491i)8-s + (0.669 + 0.743i)9-s + (0.235 − 0.971i)12-s + (0.0285 − 0.999i)13-s + (−0.935 + 0.353i)16-s + (0.999 + 0.0190i)17-s + (0.999 − 0.0380i)18-s + (0.820 + 0.572i)19-s + (−0.0475 − 0.998i)23-s + (−0.595 − 0.803i)24-s + (−0.749 − 0.662i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.245 - 0.969i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.245 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.949309420 - 2.295792381i\)
\(L(\frac12)\) \(\approx\) \(2.949309420 - 2.295792381i\)
\(L(1)\) \(\approx\) \(1.865646902 - 0.8166250181i\)
\(L(1)\) \(\approx\) \(1.865646902 - 0.8166250181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.640 - 0.768i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (0.0285 - 0.999i)T \)
17 \( 1 + (0.999 + 0.0190i)T \)
19 \( 1 + (0.820 + 0.572i)T \)
23 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.974 + 0.226i)T \)
31 \( 1 + (-0.997 - 0.0760i)T \)
37 \( 1 + (0.991 - 0.132i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.999 + 0.0380i)T \)
53 \( 1 + (0.935 + 0.353i)T \)
59 \( 1 + (0.969 + 0.244i)T \)
61 \( 1 + (0.345 - 0.938i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (-0.449 - 0.893i)T \)
79 \( 1 + (-0.953 + 0.299i)T \)
83 \( 1 + (0.254 - 0.967i)T \)
89 \( 1 + (-0.981 + 0.189i)T \)
97 \( 1 + (0.993 - 0.113i)T \)
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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

−18.584251265665302324637550139951, −17.76996050898957018776340543980, −17.104441753103598334901237258538, −16.24098148252654852979136836933, −15.7977918618974044810164456545, −14.90486798153151161795116285172, −14.47974307501263777907135982271, −13.822554244494233222642630255105, −13.3067486595617991289154791871, −12.63977344566387852288669301376, −11.84587573334008518832870969321, −11.35072302249122721855475536246, −9.98357659380946439268723115556, −9.22878370788744882259839703856, −8.83773870799814259870511377366, −7.84971684579988921697448938441, −7.2897060418408769158719210240, −6.93499707558351894995514078453, −5.78058821085794406792738417951, −5.339772134294407407979311452634, −4.01136462567231227746803290107, −3.85043620680329333376634528318, −2.82243523903981994345602152013, −2.13042674076070797655552441034, −1.03240914957028549046264793654, 0.82503101012029107010273073445, 1.70257796513571339072749327005, 2.63155344836365630585934677197, 3.17786573994097469899592468822, 3.85319534465016617422834212137, 4.56629825797787395431869917796, 5.48030140874369830899942350666, 5.92552587490443692193266007792, 7.24357533634470191893582161792, 7.84569677525941723997527031861, 8.694381215013133856687420216070, 9.52868973487594236316817313303, 9.95768952473273597784994056783, 10.68245941628039808323963985106, 11.28298805390976877223563960548, 12.3426819981079067558570744779, 12.79406923845786653771416268520, 13.429770408781287077608366571929, 14.28301087448851858898744080761, 14.66455079663425281232091084298, 15.188389255611831128023969743130, 16.13947830708963828140045646763, 16.55827938533891422242200159907, 17.87256584783521914747007251974, 18.52682261376217803137922363829

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