Properties

Label 1-3997-3997.83-r1-0-0
Degree $1$
Conductor $3997$
Sign $-0.938 + 0.345i$
Analytic cond. $429.537$
Root an. cond. $429.537$
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  + (−0.191 − 0.981i)2-s + (0.256 − 0.966i)3-s + (−0.926 + 0.376i)4-s + (−0.802 + 0.596i)5-s + (−0.997 − 0.0660i)6-s + (0.546 + 0.837i)8-s + (−0.868 − 0.495i)9-s + (0.739 + 0.673i)10-s + (0.952 + 0.303i)11-s + (0.126 + 0.991i)12-s + (0.997 + 0.0770i)13-s + (0.371 + 0.928i)15-s + (0.716 − 0.697i)16-s + (0.170 − 0.985i)17-s + (−0.319 + 0.947i)18-s + (−0.840 − 0.542i)19-s + ⋯
L(s)  = 1  + (−0.191 − 0.981i)2-s + (0.256 − 0.966i)3-s + (−0.926 + 0.376i)4-s + (−0.802 + 0.596i)5-s + (−0.997 − 0.0660i)6-s + (0.546 + 0.837i)8-s + (−0.868 − 0.495i)9-s + (0.739 + 0.673i)10-s + (0.952 + 0.303i)11-s + (0.126 + 0.991i)12-s + (0.997 + 0.0770i)13-s + (0.371 + 0.928i)15-s + (0.716 − 0.697i)16-s + (0.170 − 0.985i)17-s + (−0.319 + 0.947i)18-s + (−0.840 − 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $-0.938 + 0.345i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ -0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2635861914 - 1.480503984i\)
\(L(\frac12)\) \(\approx\) \(-0.2635861914 - 1.480503984i\)
\(L(1)\) \(\approx\) \(0.6139387208 - 0.6303330246i\)
\(L(1)\) \(\approx\) \(0.6139387208 - 0.6303330246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.191 - 0.981i)T \)
3 \( 1 + (0.256 - 0.966i)T \)
5 \( 1 + (-0.802 + 0.596i)T \)
11 \( 1 + (0.952 + 0.303i)T \)
13 \( 1 + (0.997 + 0.0770i)T \)
17 \( 1 + (0.170 - 0.985i)T \)
19 \( 1 + (-0.840 - 0.542i)T \)
23 \( 1 + (-0.627 - 0.778i)T \)
29 \( 1 + (0.0275 + 0.999i)T \)
31 \( 1 + (-0.245 - 0.969i)T \)
37 \( 1 + (-0.381 - 0.924i)T \)
41 \( 1 + (0.298 - 0.954i)T \)
43 \( 1 + (0.202 + 0.979i)T \)
47 \( 1 + (0.962 + 0.272i)T \)
53 \( 1 + (-0.421 - 0.906i)T \)
59 \( 1 + (0.986 - 0.164i)T \)
61 \( 1 + (-0.391 - 0.920i)T \)
67 \( 1 + (0.996 + 0.0880i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.609 - 0.792i)T \)
79 \( 1 + (0.984 + 0.175i)T \)
83 \( 1 + (0.556 - 0.831i)T \)
89 \( 1 + (0.441 + 0.897i)T \)
97 \( 1 + (-0.970 - 0.240i)T \)
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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

−18.829790418388124390989202125199, −17.71826998444211732177233157298, −16.933296587336251696572999932852, −16.70069875727386797469332523286, −15.90175824844279652318543876929, −15.37717108229280826781501336521, −14.92999453335966071553917684801, −14.05937876839053759126808858270, −13.56011951818409387888072251158, −12.59099063143290759845659994465, −11.80835422062043553717178249127, −10.93651431087438607421434461907, −10.29520886649593628699239625014, −9.43508723360403724761308701127, −8.78730214558502593684150060204, −8.30660332645713918439232497398, −7.83421686061702413370139176932, −6.65164342787228829323856017742, −5.94282114724086571229430434334, −5.32746352081998685795761660631, −4.26423432300001956126539883489, −3.960908144936938770129367503600, −3.38729905134346756481832051480, −1.70829949376303499063977982413, −0.80983678349882735901070759072, 0.35694988625376061953413396166, 0.895859519905136946529832756528, 2.01907260636562353272147625476, 2.520705638531664147535037797021, 3.5727944040847512073915295349, 3.8973114211603251865095593857, 4.93354521721059465451618169729, 6.13211290881986286859521831805, 6.81008542240723586152189075807, 7.52062616336859887954378074847, 8.255763566658065759078247129793, 8.891981859149439525357446016364, 9.48581756171152147488070905654, 10.64404642028153902347179306851, 11.19276232954727817591808083853, 11.6860023269160035189233629318, 12.461632727567993428939857892090, 12.84221890466013336473447716504, 13.93087226367993821660053833885, 14.21636657256816421883502361121, 14.934621960191148667142601634770, 15.975881159118835309562885276173, 16.76459818980753711966226976016, 17.665846528995241384289513145050, 18.1288847178827527785970050064

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