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Properties

Label 1-4235-4235.1997-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.856 - 0.515i$
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  + (0.836 − 0.548i)2-s + (−0.207 + 0.978i)3-s + (0.398 − 0.917i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (−0.913 − 0.406i)9-s + (0.814 + 0.580i)12-s + (0.491 − 0.870i)13-s + (−0.683 − 0.730i)16-s + (0.647 + 0.761i)17-s + (−0.986 + 0.161i)18-s + (−0.851 + 0.524i)19-s + (0.189 − 0.981i)23-s + (0.999 + 0.0380i)24-s + (−0.0665 − 0.997i)26-s + (0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (0.836 − 0.548i)2-s + (−0.207 + 0.978i)3-s + (0.398 − 0.917i)4-s + (0.362 + 0.931i)6-s + (−0.170 − 0.985i)8-s + (−0.913 − 0.406i)9-s + (0.814 + 0.580i)12-s + (0.491 − 0.870i)13-s + (−0.683 − 0.730i)16-s + (0.647 + 0.761i)17-s + (−0.986 + 0.161i)18-s + (−0.851 + 0.524i)19-s + (0.189 − 0.981i)23-s + (0.999 + 0.0380i)24-s + (−0.0665 − 0.997i)26-s + (0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.856 - 0.515i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1997, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.856 - 0.515i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3176157697 - 1.143945765i\)
\(L(\frac12)\) \(\approx\) \(0.3176157697 - 1.143945765i\)
\(L(1)\) \(\approx\) \(1.255132093 - 0.3170304157i\)
\(L(1)\) \(\approx\) \(1.255132093 - 0.3170304157i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.836 - 0.548i)T \)
3 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 + (0.491 - 0.870i)T \)
17 \( 1 + (0.647 + 0.761i)T \)
19 \( 1 + (-0.851 + 0.524i)T \)
23 \( 1 + (0.189 - 0.981i)T \)
29 \( 1 + (-0.564 + 0.825i)T \)
31 \( 1 + (-0.948 - 0.318i)T \)
37 \( 1 + (-0.976 - 0.217i)T \)
41 \( 1 + (0.254 - 0.967i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 + (0.986 + 0.161i)T \)
53 \( 1 + (0.730 + 0.683i)T \)
59 \( 1 + (-0.964 - 0.263i)T \)
61 \( 1 + (-0.449 - 0.893i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (-0.0190 - 0.999i)T \)
79 \( 1 + (0.272 - 0.962i)T \)
83 \( 1 + (-0.996 + 0.0855i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
97 \( 1 + (0.884 - 0.466i)T \)
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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

−18.49906947636792288476875834173, −17.97363743698640372809580538139, −17.049770974626183944979601406357, −16.75284217738632923639536007164, −15.96113698420583599009760164912, −15.143295730413930449331767620674, −14.491794431882727465239378887334, −13.6656195129761736207899304834, −13.454608324080239964279726584246, −12.63667653075449492145600000624, −11.92761643390005115826519223539, −11.43987704243169130822018089641, −10.81796690831825382658510247962, −9.51033333738007444863995830189, −8.70287450049825221662861002830, −8.07902161966794789296228341315, −7.10095822209539686404852535259, −6.99023764217523233733244253742, −5.974487963807618107907479181795, −5.49453234262878003751220386390, −4.64402946386019462201748245302, −3.758283467027459020180049881061, −2.94238191321088806564412811230, −2.11236602304010913515484768614, −1.32756367354634790634870436536, 0.22945187539572877497141696726, 1.46192966425304094339509395749, 2.39091093862917616776926222716, 3.424996643033032536524943110958, 3.69555299757121664225232277511, 4.56525489125322424039578873123, 5.3587805825960954619747810611, 5.85466341441595210754651168067, 6.53536769452523134610638148616, 7.64518672504755866009580523367, 8.65123427146104709963033757227, 9.22084005111337620772611219805, 10.31379141357190443763455948929, 10.54982306740695764053784432485, 11.028031209816379444058724060924, 12.114373956596175258631637711, 12.50663560097247892663194965076, 13.2463344407705439624272469735, 14.23426828281570213523203624842, 14.64486335226639876689221130661, 15.31469743742159123215484269902, 15.817089039047216743250858272396, 16.73041866954726215043758536808, 17.12113793206694855871221452983, 18.324620047965982619773345841797

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