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Properties

Label 2-201-67.66-c4-0-18
Degree $2$
Conductor $201$
Sign $0.547 - 0.836i$
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  − 0.364i·2-s + 5.19i·3-s + 15.8·4-s − 25.2i·5-s + 1.89·6-s + 55.9i·7-s − 11.6i·8-s − 27·9-s − 9.18·10-s + 6.35i·11-s + 82.4i·12-s + 237. i·13-s + 20.4·14-s + 130.·15-s + 249.·16-s + 123.·17-s + ⋯
L(s)  = 1  − 0.0911i·2-s + 0.577i·3-s + 0.991·4-s − 1.00i·5-s + 0.0526·6-s + 1.14i·7-s − 0.181i·8-s − 0.333·9-s − 0.0918·10-s + 0.0525i·11-s + 0.572i·12-s + 1.40i·13-s + 0.104·14-s + 0.582·15-s + 0.975·16-s + 0.428·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.547 - 0.836i$
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: \((2,\ 201,\ (\ :2),\ 0.547 - 0.836i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.344024851\)
\(L(\frac12)\) \(\approx\) \(2.344024851\)
\(L(3)\) 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 \( 1 + (3.75e3 + 2.45e3i)T \)
good2 \( 1 + 0.364iT - 16T^{2} \)
5 \( 1 + 25.2iT - 625T^{2} \)
7 \( 1 - 55.9iT - 2.40e3T^{2} \)
11 \( 1 - 6.35iT - 1.46e4T^{2} \)
13 \( 1 - 237. iT - 2.85e4T^{2} \)
17 \( 1 - 123.T + 8.35e4T^{2} \)
19 \( 1 - 127.T + 1.30e5T^{2} \)
23 \( 1 + 366.T + 2.79e5T^{2} \)
29 \( 1 - 1.03e3T + 7.07e5T^{2} \)
31 \( 1 - 718. iT - 9.23e5T^{2} \)
37 \( 1 - 53.2T + 1.87e6T^{2} \)
41 \( 1 - 1.36e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.66e3iT - 3.41e6T^{2} \)
47 \( 1 - 602.T + 4.87e6T^{2} \)
53 \( 1 - 3.27e3iT - 7.89e6T^{2} \)
59 \( 1 - 827.T + 1.21e7T^{2} \)
61 \( 1 + 2.48e3iT - 1.38e7T^{2} \)
71 \( 1 - 3.71e3T + 2.54e7T^{2} \)
73 \( 1 - 1.99e3T + 2.83e7T^{2} \)
79 \( 1 + 1.30e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.02e3T + 4.74e7T^{2} \)
89 \( 1 + 3.40e3T + 6.27e7T^{2} \)
97 \( 1 - 6.07e3iT - 8.85e7T^{2} \)
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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

−12.05977441755620758400456686416, −11.12340341365836719019167392740, −9.863823353242019378809340384843, −9.027260434766351244083805540497, −8.113475893665903037437478272925, −6.61463947426406150057692363015, −5.57972174221620831349640784676, −4.46482733371987803600227600757, −2.84927247961200081107466906388, −1.50485380711188792290275353801, 0.866572919309475251382942244175, 2.52911455076598682718488041033, 3.56300960517320256348468365469, 5.62293433684091365607302997630, 6.68544838051776177514951883561, 7.39233981820021585561057672489, 8.122976164920649361059343371629, 10.19523844154520307087458072197, 10.55151130939415700102501547617, 11.55214391089785854228646002303

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