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Properties

Label 1-4235-4235.488-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.590 + 0.807i$
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  + (−0.917 + 0.398i)2-s + (−0.406 − 0.913i)3-s + (0.683 − 0.730i)4-s + (0.736 + 0.676i)6-s + (−0.336 + 0.941i)8-s + (−0.669 + 0.743i)9-s + (−0.945 − 0.327i)12-s + (−0.856 − 0.516i)13-s + (−0.0665 − 0.997i)16-s + (−0.986 + 0.161i)17-s + (0.318 − 0.948i)18-s + (0.449 + 0.893i)19-s + (0.371 + 0.928i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.917 + 0.398i)2-s + (−0.406 − 0.913i)3-s + (0.683 − 0.730i)4-s + (0.736 + 0.676i)6-s + (−0.336 + 0.941i)8-s + (−0.669 + 0.743i)9-s + (−0.945 − 0.327i)12-s + (−0.856 − 0.516i)13-s + (−0.0665 − 0.997i)16-s + (−0.986 + 0.161i)17-s + (0.318 − 0.948i)18-s + (0.449 + 0.893i)19-s + (0.371 + 0.928i)23-s + (0.997 − 0.0760i)24-s + (0.991 + 0.132i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.590 + 0.807i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.590 + 0.807i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07321878742 + 0.1443028136i\)
\(L(\frac12)\) \(\approx\) \(0.07321878742 + 0.1443028136i\)
\(L(1)\) \(\approx\) \(0.5087170920 - 0.04939659195i\)
\(L(1)\) \(\approx\) \(0.5087170920 - 0.04939659195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.917 + 0.398i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.856 - 0.516i)T \)
17 \( 1 + (-0.986 + 0.161i)T \)
19 \( 1 + (0.449 + 0.893i)T \)
23 \( 1 + (0.371 + 0.928i)T \)
29 \( 1 + (0.362 - 0.931i)T \)
31 \( 1 + (-0.797 + 0.603i)T \)
37 \( 1 + (0.424 + 0.905i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (-0.318 - 0.948i)T \)
53 \( 1 + (-0.997 - 0.0665i)T \)
59 \( 1 + (0.861 - 0.508i)T \)
61 \( 1 + (0.595 + 0.803i)T \)
67 \( 1 + (0.814 - 0.580i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (0.0380 - 0.999i)T \)
79 \( 1 + (0.851 - 0.524i)T \)
83 \( 1 + (-0.170 + 0.985i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.825 - 0.564i)T \)
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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

−17.92712373693884204354218412880, −17.53827319554893925002725820141, −16.8451339205488787727346585115, −16.085074430545217356310701155329, −15.876488837987205777248775992098, −14.79421339742976383764139085546, −14.40039114338461496386882689712, −13.0648723844300732141249840616, −12.55483381302096282446513119084, −11.62327060293661989383838142364, −11.14129216527605763629492097176, −10.69119795942012710751107415640, −9.680909422245528840581719457818, −9.36758860097007304352329350703, −8.749354928512019103383913794360, −7.85718893523595580352605976108, −6.902043021489720810400773555923, −6.48936656575209068530839204394, −5.349978379442319316278850651728, −4.55360716955828367289102074953, −3.942520239940693288140262865675, −2.826864749705411960212209338605, −2.42132717594221552532692573949, −1.11764142245628810727354471482, −0.08220467835396216815207439946, 0.91193628455782364192280442015, 1.8401516318919791259426810000, 2.41080167025169316638576028467, 3.44563264503685984640030138254, 4.87763665278660473070558913904, 5.45220054834833124332596697166, 6.203255998369231153697179721907, 6.875824887624029800423094362662, 7.538278487726339840424422777065, 8.041693890630412833581264245319, 8.807327886831582821581204709045, 9.63538975277022369171874618897, 10.3431128160447187052743715447, 11.05188004939218037462401544865, 11.73840882049586712039640740212, 12.306216379674645044658407283488, 13.18648719396887719318666268981, 13.87609357311918735649086021751, 14.6654647338320627851132866994, 15.32837386541100731459212101223, 16.090878941788693741498801719398, 16.80456405533970313092491962721, 17.43387784257340093513834585068, 17.80042700680564521859803090995, 18.51356141435776281253063390416

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