Properties

Label 2-201-67.66-c4-0-1
Degree $2$
Conductor $201$
Sign $0.908 - 0.418i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.56i·2-s − 5.19i·3-s − 15.0·4-s − 37.5i·5-s − 28.9·6-s + 91.3i·7-s − 5.54i·8-s − 27·9-s − 208.·10-s + 134. i·11-s + 77.9i·12-s + 280. i·13-s + 508.·14-s − 194.·15-s − 270.·16-s + 68.0·17-s + ⋯
L(s)  = 1  − 1.39i·2-s − 0.577i·3-s − 0.937·4-s − 1.50i·5-s − 0.803·6-s + 1.86i·7-s − 0.0866i·8-s − 0.333·9-s − 2.08·10-s + 1.10i·11-s + 0.541i·12-s + 1.66i·13-s + 2.59·14-s − 0.866·15-s − 1.05·16-s + 0.235·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.908 - 0.418i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ 0.908 - 0.418i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4216089232\)
\(L(\frac12)\) \(\approx\) \(0.4216089232\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (-1.87e3 - 4.07e3i)T \)
good2 \( 1 + 5.56iT - 16T^{2} \)
5 \( 1 + 37.5iT - 625T^{2} \)
7 \( 1 - 91.3iT - 2.40e3T^{2} \)
11 \( 1 - 134. iT - 1.46e4T^{2} \)
13 \( 1 - 280. iT - 2.85e4T^{2} \)
17 \( 1 - 68.0T + 8.35e4T^{2} \)
19 \( 1 + 402.T + 1.30e5T^{2} \)
23 \( 1 + 614.T + 2.79e5T^{2} \)
29 \( 1 + 861.T + 7.07e5T^{2} \)
31 \( 1 - 485. iT - 9.23e5T^{2} \)
37 \( 1 - 608.T + 1.87e6T^{2} \)
41 \( 1 + 2.94e3iT - 2.82e6T^{2} \)
43 \( 1 - 789. iT - 3.41e6T^{2} \)
47 \( 1 + 1.77e3T + 4.87e6T^{2} \)
53 \( 1 + 3.28e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.16e3T + 1.21e7T^{2} \)
61 \( 1 + 5.82e3iT - 1.38e7T^{2} \)
71 \( 1 + 2.07e3T + 2.54e7T^{2} \)
73 \( 1 - 9.34e3T + 2.83e7T^{2} \)
79 \( 1 - 3.63e3iT - 3.89e7T^{2} \)
83 \( 1 + 7.67e3T + 4.74e7T^{2} \)
89 \( 1 - 1.93e3T + 6.27e7T^{2} \)
97 \( 1 - 441. iT - 8.85e7T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13870829923112747007385436037, −11.33885618174359596092928777581, −9.639488811119361495440920229722, −9.143055856672041342578807637639, −8.312949856476305108918222822024, −6.55627566277175744220946158339, −5.20395406096576107401846144893, −4.12465969872749596965701144093, −2.08548021587061852182799493917, −1.81604679223602963740283015183, 0.13844994263227391295044718047, 3.09529454987821633501485315834, 4.16791265309237345211312057013, 5.78693453833987800416108269568, 6.53137652556175542302418045973, 7.62950577303141935869801050629, 8.114684311770490802093009736781, 9.904132212222727280062757758959, 10.72757032707186178961544375348, 11.13640849539802183654968051827

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