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Properties

Label 1-3997-3997.202-r1-0-0
Degree $1$
Conductor $3997$
Sign $0.858 - 0.512i$
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  + (−0.821 + 0.569i)2-s + (−0.618 − 0.785i)3-s + (0.350 − 0.936i)4-s + (0.256 + 0.966i)5-s + (0.956 + 0.293i)6-s + (0.245 + 0.969i)8-s + (−0.234 + 0.972i)9-s + (−0.761 − 0.648i)10-s + (−0.942 − 0.335i)11-s + (−0.952 + 0.303i)12-s + (0.644 + 0.764i)13-s + (0.601 − 0.799i)15-s + (−0.754 − 0.656i)16-s + (0.857 − 0.514i)17-s + (−0.360 − 0.932i)18-s + (0.0385 + 0.999i)19-s + ⋯
L(s)  = 1  + (−0.821 + 0.569i)2-s + (−0.618 − 0.785i)3-s + (0.350 − 0.936i)4-s + (0.256 + 0.966i)5-s + (0.956 + 0.293i)6-s + (0.245 + 0.969i)8-s + (−0.234 + 0.972i)9-s + (−0.761 − 0.648i)10-s + (−0.942 − 0.335i)11-s + (−0.952 + 0.303i)12-s + (0.644 + 0.764i)13-s + (0.601 − 0.799i)15-s + (−0.754 − 0.656i)16-s + (0.857 − 0.514i)17-s + (−0.360 − 0.932i)18-s + (0.0385 + 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3997 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3997\)    =    \(7 \cdot 571\)
Sign: $0.858 - 0.512i$
Analytic conductor: \(429.537\)
Root analytic conductor: \(429.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3997} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3997,\ (1:\ ),\ 0.858 - 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6073815405 - 0.1675665957i\)
\(L(\frac12)\) \(\approx\) \(0.6073815405 - 0.1675665957i\)
\(L(1)\) \(\approx\) \(0.5562702456 + 0.1090057792i\)
\(L(1)\) \(\approx\) \(0.5562702456 + 0.1090057792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
571 \( 1 \)
good2 \( 1 + (-0.821 + 0.569i)T \)
3 \( 1 + (-0.618 - 0.785i)T \)
5 \( 1 + (0.256 + 0.966i)T \)
11 \( 1 + (-0.942 - 0.335i)T \)
13 \( 1 + (0.644 + 0.764i)T \)
17 \( 1 + (0.857 - 0.514i)T \)
19 \( 1 + (0.0385 + 0.999i)T \)
23 \( 1 + (-0.768 + 0.639i)T \)
29 \( 1 + (0.137 - 0.990i)T \)
31 \( 1 + (-0.945 + 0.324i)T \)
37 \( 1 + (0.0715 - 0.997i)T \)
41 \( 1 + (0.998 + 0.0550i)T \)
43 \( 1 + (-0.381 + 0.924i)T \)
47 \( 1 + (0.191 - 0.981i)T \)
53 \( 1 + (-0.795 + 0.605i)T \)
59 \( 1 + (0.677 + 0.735i)T \)
61 \( 1 + (0.480 + 0.876i)T \)
67 \( 1 + (-0.126 - 0.991i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.899 - 0.436i)T \)
79 \( 1 + (-0.968 + 0.250i)T \)
83 \( 1 + (-0.731 + 0.681i)T \)
89 \( 1 + (-0.224 - 0.974i)T \)
97 \( 1 + (-0.999 - 0.0440i)T \)
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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

−18.16728225681060413729813749072, −17.67885288161618628721315686731, −17.11863838396917940245446242985, −16.383038080640738206348638167, −15.90361993687420517653802814778, −15.41228138727646129378150622918, −14.33695172182554496140375121492, −13.18139126071338791099068466838, −12.673786471443954235642221536687, −12.23944243766903323270092644007, −11.194708494683513290985752219133, −10.79660763447310525552949886265, −9.922793427351413386439447645735, −9.67561550369131878946790718692, −8.60649520417414152569943153652, −8.30413470749077801940104814815, −7.343843416027725361733739897904, −6.31310324733029765982356362989, −5.505021143738147632040844948947, −4.879859678535007924724254463567, −4.01407505863762175043301776989, −3.27228607468943852998301221751, −2.33843119492953700488390405757, −1.26411160668516360644890458119, −0.558066799910994838309955981102, 0.23106934085027385967317561624, 1.31131166219526635197540226619, 2.017313591283155592748822839929, 2.77746287700576870934012250154, 3.93990703919686028561392731804, 5.299196684811759524846758117453, 5.79919728909383510230615950842, 6.28372530831962635826470584774, 7.13394827407454148972840332708, 7.70756418255647612614200426257, 8.14415126292243439324504285390, 9.2715536849008512291463170366, 10.04063669713219146387951263725, 10.61145263600449846247938018387, 11.2935081649067575315485331555, 11.779924794877855936604833247423, 12.80249400818959963696692026400, 13.88192822334060588132944490348, 13.97530752686343216991036754228, 14.87084721699239552782656896493, 15.86694273604908975683638037700, 16.28602309685192565520325208965, 16.9340847885829493900338159276, 17.82725230199010565838985765191, 18.314048476079394654969475396465

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