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Properties

Label 2-201-67.66-c4-0-6
Degree $2$
Conductor $201$
Sign $-0.692 + 0.721i$
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  + 4.05i·2-s + 5.19i·3-s − 0.433·4-s − 2.19i·5-s − 21.0·6-s − 19.8i·7-s + 63.1i·8-s − 27·9-s + 8.89·10-s − 61.3i·11-s − 2.25i·12-s + 236. i·13-s + 80.4·14-s + 11.3·15-s − 262.·16-s − 483.·17-s + ⋯
L(s)  = 1  + 1.01i·2-s + 0.577i·3-s − 0.0271·4-s − 0.0877i·5-s − 0.585·6-s − 0.405i·7-s + 0.985i·8-s − 0.333·9-s + 0.0889·10-s − 0.507i·11-s − 0.0156i·12-s + 1.39i·13-s + 0.410·14-s + 0.0506·15-s − 1.02·16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.692 + 0.721i$
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: \((2,\ 201,\ (\ :2),\ -0.692 + 0.721i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9793520229\)
\(L(\frac12)\) \(\approx\) \(0.9793520229\)
\(L(3)\) 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 \( 1 + (-3.23e3 - 3.11e3i)T \)
good2 \( 1 - 4.05iT - 16T^{2} \)
5 \( 1 + 2.19iT - 625T^{2} \)
7 \( 1 + 19.8iT - 2.40e3T^{2} \)
11 \( 1 + 61.3iT - 1.46e4T^{2} \)
13 \( 1 - 236. iT - 2.85e4T^{2} \)
17 \( 1 + 483.T + 8.35e4T^{2} \)
19 \( 1 + 407.T + 1.30e5T^{2} \)
23 \( 1 + 331.T + 2.79e5T^{2} \)
29 \( 1 + 174.T + 7.07e5T^{2} \)
31 \( 1 - 1.55e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.37e3T + 1.87e6T^{2} \)
41 \( 1 + 1.73e3iT - 2.82e6T^{2} \)
43 \( 1 + 806. iT - 3.41e6T^{2} \)
47 \( 1 - 1.72e3T + 4.87e6T^{2} \)
53 \( 1 + 491. iT - 7.89e6T^{2} \)
59 \( 1 + 5.49e3T + 1.21e7T^{2} \)
61 \( 1 + 3.38e3iT - 1.38e7T^{2} \)
71 \( 1 - 465.T + 2.54e7T^{2} \)
73 \( 1 - 8.58e3T + 2.83e7T^{2} \)
79 \( 1 - 1.07e4iT - 3.89e7T^{2} \)
83 \( 1 + 563.T + 4.74e7T^{2} \)
89 \( 1 + 1.24e4T + 6.27e7T^{2} \)
97 \( 1 + 1.09e4iT - 8.85e7T^{2} \)
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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

−12.36296721355076169524433155336, −11.15410706446649201864901716557, −10.62400397879872583920467409824, −8.981878409976849509091435610615, −8.543987426295072568887350281091, −7.00140692509267927750229104839, −6.44092655165764208025133878230, −5.06442849562205385864220973713, −4.03421340983585001267922823019, −2.16598166131246077072535212840, 0.31155571877504415767531514674, 1.94477919277977985204391022849, 2.81922132778814189912031917052, 4.34883963891471718536162401066, 6.00765310970726287503656388272, 6.98777673398405635214015143888, 8.207881341746077680388454038704, 9.345885623803106551461390928517, 10.53288187578698661976254853990, 11.11939078689879636515259628748

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