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Properties

Label 2-201-67.30-c2-0-4
Degree $2$
Conductor $201$
Sign $-0.0665 - 0.997i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.15 − 0.668i)2-s + 1.73i·3-s + (−1.10 + 1.91i)4-s + 0.749i·5-s + (1.15 + 2.00i)6-s + (2.89 + 1.67i)7-s + 8.30i·8-s − 2.99·9-s + (0.501 + 0.867i)10-s + (−6.87 − 3.96i)11-s + (−3.31 − 1.91i)12-s + (−14.8 + 8.59i)13-s + 4.46·14-s − 1.29·15-s + (1.12 + 1.95i)16-s + (9.50 + 16.4i)17-s + ⋯
L(s)  = 1  + (0.578 − 0.334i)2-s + 0.577i·3-s + (−0.276 + 0.478i)4-s + 0.149i·5-s + (0.192 + 0.334i)6-s + (0.413 + 0.238i)7-s + 1.03i·8-s − 0.333·9-s + (0.0501 + 0.0867i)10-s + (−0.625 − 0.360i)11-s + (−0.276 − 0.159i)12-s + (−1.14 + 0.661i)13-s + 0.319·14-s − 0.0865·15-s + (0.0705 + 0.122i)16-s + (0.558 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0665 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0665 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0665 - 0.997i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.0665 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13254 + 1.21056i\)
\(L(\frac12)\) \(\approx\) \(1.13254 + 1.21056i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
67 \( 1 + (-47.9 - 46.8i)T \)
good2 \( 1 + (-1.15 + 0.668i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 0.749iT - 25T^{2} \)
7 \( 1 + (-2.89 - 1.67i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (6.87 + 3.96i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (14.8 - 8.59i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-9.50 - 16.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 - 25.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.701 + 1.21i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-26.1 + 45.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (16.5 + 9.53i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (6.29 + 10.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-46.7 - 26.9i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 - 7.92iT - 1.84e3T^{2} \)
47 \( 1 + (-16.8 + 29.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 26.7iT - 2.80e3T^{2} \)
59 \( 1 - 106.T + 3.48e3T^{2} \)
61 \( 1 + (59.8 - 34.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-15.3 + 26.6i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (56.9 + 98.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-54.9 - 31.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (8.76 + 15.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 123.T + 7.92e3T^{2} \)
97 \( 1 + (119. - 68.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.30873221102940519596862911141, −11.74321050674383985108297366160, −10.61970311150214151047771888301, −9.647857714769291498367880268385, −8.411127434583656658156327090860, −7.64518656093061538650318156467, −5.81658736901264522497422505673, −4.82069442420667801434253974609, −3.74587213065125292485071162691, −2.45746857469612742378394831034, 0.820012562782244115629837972089, 2.88250283234480075270936392794, 4.89662156700058680221784747530, 5.26434562205415538463795558745, 6.91518995517571632932808768310, 7.52239741952286627623656752160, 8.993237947710588749708645872683, 10.00533334227247752988800557811, 11.02774308516546738567500941529, 12.37789494289712649339151958724

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