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Properties

Label 2-201-67.66-c4-0-25
Degree $2$
Conductor $201$
Sign $-0.0847 + 0.996i$
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  − 5.76i·2-s + 5.19i·3-s − 17.1·4-s − 16.3i·5-s + 29.9·6-s + 86.3i·7-s + 6.81i·8-s − 27·9-s − 94.0·10-s + 86.2i·11-s − 89.2i·12-s − 311. i·13-s + 497.·14-s + 84.7·15-s − 235.·16-s + 434.·17-s + ⋯
L(s)  = 1  − 1.44i·2-s + 0.577i·3-s − 1.07·4-s − 0.652i·5-s + 0.831·6-s + 1.76i·7-s + 0.106i·8-s − 0.333·9-s − 0.940·10-s + 0.712i·11-s − 0.620i·12-s − 1.84i·13-s + 2.53·14-s + 0.376·15-s − 0.920·16-s + 1.50·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0847 + 0.996i$
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: \((2,\ 201,\ (\ :2),\ -0.0847 + 0.996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.888271985\)
\(L(\frac12)\) \(\approx\) \(1.888271985\)
\(L(3)\) 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 \( 1 + (-4.47e3 - 380. i)T \)
good2 \( 1 + 5.76iT - 16T^{2} \)
5 \( 1 + 16.3iT - 625T^{2} \)
7 \( 1 - 86.3iT - 2.40e3T^{2} \)
11 \( 1 - 86.2iT - 1.46e4T^{2} \)
13 \( 1 + 311. iT - 2.85e4T^{2} \)
17 \( 1 - 434.T + 8.35e4T^{2} \)
19 \( 1 - 485.T + 1.30e5T^{2} \)
23 \( 1 + 180.T + 2.79e5T^{2} \)
29 \( 1 - 1.18e3T + 7.07e5T^{2} \)
31 \( 1 + 1.41e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.09e3T + 1.87e6T^{2} \)
41 \( 1 + 814. iT - 2.82e6T^{2} \)
43 \( 1 + 919. iT - 3.41e6T^{2} \)
47 \( 1 - 2.94e3T + 4.87e6T^{2} \)
53 \( 1 - 1.77e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.86e3T + 1.21e7T^{2} \)
61 \( 1 + 2.31e3iT - 1.38e7T^{2} \)
71 \( 1 - 6.04e3T + 2.54e7T^{2} \)
73 \( 1 + 5.03e3T + 2.83e7T^{2} \)
79 \( 1 - 5.51e3iT - 3.89e7T^{2} \)
83 \( 1 + 2.71e3T + 4.74e7T^{2} \)
89 \( 1 - 7.75e3T + 6.27e7T^{2} \)
97 \( 1 + 695. iT - 8.85e7T^{2} \)
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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

−11.79538345178694622656933892239, −10.43884830434666158354209172461, −9.774164094958776105082759001844, −8.967169283051082333342815762274, −7.898055618526187500154441847192, −5.68508032906538807304651916703, −5.00360693477367655500935373229, −3.36769416337185956479381487359, −2.49149668908218856106176558052, −0.868054226998291807168774843598, 1.10678817018813291890416017156, 3.38120082169039906081877204521, 4.82688081121907543601646299791, 6.25787161492816670909867736004, 7.03398553614658864575143026859, 7.49099118592681813235314941104, 8.614839756758779420032201177888, 9.975847355310161100633432675676, 11.05498185111941121080738554068, 12.04183621405331123747581007400

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