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Properties

Label 2-201-67.66-c4-0-26
Degree $2$
Conductor $201$
Sign $0.283 - 0.958i$
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  + 7.50i·2-s + 5.19i·3-s − 40.3·4-s − 11.1i·5-s − 39.0·6-s − 57.6i·7-s − 183. i·8-s − 27·9-s + 83.7·10-s + 61.5i·11-s − 209. i·12-s + 14.9i·13-s + 432.·14-s + 57.9·15-s + 728.·16-s − 231.·17-s + ⋯
L(s)  = 1  + 1.87i·2-s + 0.577i·3-s − 2.52·4-s − 0.446i·5-s − 1.08·6-s − 1.17i·7-s − 2.86i·8-s − 0.333·9-s + 0.837·10-s + 0.508i·11-s − 1.45i·12-s + 0.0887i·13-s + 2.20·14-s + 0.257·15-s + 2.84·16-s − 0.801·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.283 - 0.958i$
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: \((2,\ 201,\ (\ :2),\ 0.283 - 0.958i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.264746799\)
\(L(\frac12)\) \(\approx\) \(1.264746799\)
\(L(3)\) 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 \( 1 + (4.30e3 + 1.27e3i)T \)
good2 \( 1 - 7.50iT - 16T^{2} \)
5 \( 1 + 11.1iT - 625T^{2} \)
7 \( 1 + 57.6iT - 2.40e3T^{2} \)
11 \( 1 - 61.5iT - 1.46e4T^{2} \)
13 \( 1 - 14.9iT - 2.85e4T^{2} \)
17 \( 1 + 231.T + 8.35e4T^{2} \)
19 \( 1 - 627.T + 1.30e5T^{2} \)
23 \( 1 + 308.T + 2.79e5T^{2} \)
29 \( 1 + 73.3T + 7.07e5T^{2} \)
31 \( 1 + 368. iT - 9.23e5T^{2} \)
37 \( 1 - 1.52e3T + 1.87e6T^{2} \)
41 \( 1 + 2.57e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.63e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.22e3T + 4.87e6T^{2} \)
53 \( 1 - 1.13e3iT - 7.89e6T^{2} \)
59 \( 1 + 4.02e3T + 1.21e7T^{2} \)
61 \( 1 - 1.74e3iT - 1.38e7T^{2} \)
71 \( 1 - 4.69e3T + 2.54e7T^{2} \)
73 \( 1 + 697.T + 2.83e7T^{2} \)
79 \( 1 + 1.71e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.00e4T + 4.74e7T^{2} \)
89 \( 1 + 193.T + 6.27e7T^{2} \)
97 \( 1 - 3.60e3iT - 8.85e7T^{2} \)
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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

−12.26985274305798743273117922124, −10.63197950733095642440534169373, −9.591737095462848737365264301476, −8.853784521292505526643282809699, −7.64140121631890572029118598131, −7.02492587893605553910799995898, −5.70539656759763457287976116839, −4.70502917257192104202285055779, −3.89505862074565382596563307783, −0.56166587255677454877344909795, 1.10735861664721990059055405377, 2.48187463760801818805241517091, 3.20323263834204293595384359999, 4.86846554811769150296890645756, 6.10300047495636432453321338343, 7.922181020590115599996653692709, 8.974440862322451875215570709058, 9.711207847710067798717023452522, 10.98679148007927206447522135986, 11.56272828280518170250921344298

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