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Properties

Label 2-201-67.66-c4-0-15
Degree $2$
Conductor $201$
Sign $-0.950 - 0.311i$
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  + 1.39i·2-s + 5.19i·3-s + 14.0·4-s + 17.8i·5-s − 7.25·6-s + 70.8i·7-s + 41.9i·8-s − 27·9-s − 24.8·10-s + 59.5i·11-s + 73.0i·12-s − 138. i·13-s − 98.8·14-s − 92.5·15-s + 166.·16-s − 130.·17-s + ⋯
L(s)  = 1  + 0.348i·2-s + 0.577i·3-s + 0.878·4-s + 0.712i·5-s − 0.201·6-s + 1.44i·7-s + 0.655i·8-s − 0.333·9-s − 0.248·10-s + 0.492i·11-s + 0.507i·12-s − 0.819i·13-s − 0.504·14-s − 0.411·15-s + 0.649·16-s − 0.452·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.950 - 0.311i$
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: \((2,\ 201,\ (\ :2),\ -0.950 - 0.311i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.033749774\)
\(L(\frac12)\) \(\approx\) \(2.033749774\)
\(L(3)\) 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 \( 1 + (1.39e3 - 4.26e3i)T \)
good2 \( 1 - 1.39iT - 16T^{2} \)
5 \( 1 - 17.8iT - 625T^{2} \)
7 \( 1 - 70.8iT - 2.40e3T^{2} \)
11 \( 1 - 59.5iT - 1.46e4T^{2} \)
13 \( 1 + 138. iT - 2.85e4T^{2} \)
17 \( 1 + 130.T + 8.35e4T^{2} \)
19 \( 1 + 343.T + 1.30e5T^{2} \)
23 \( 1 - 731.T + 2.79e5T^{2} \)
29 \( 1 + 368.T + 7.07e5T^{2} \)
31 \( 1 + 574. iT - 9.23e5T^{2} \)
37 \( 1 + 425.T + 1.87e6T^{2} \)
41 \( 1 - 2.26e3iT - 2.82e6T^{2} \)
43 \( 1 + 3.23e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.95e3T + 4.87e6T^{2} \)
53 \( 1 - 1.57e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.48e3T + 1.21e7T^{2} \)
61 \( 1 - 3.76e3iT - 1.38e7T^{2} \)
71 \( 1 - 2.99e3T + 2.54e7T^{2} \)
73 \( 1 - 8.19e3T + 2.83e7T^{2} \)
79 \( 1 + 2.42e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.69e3T + 4.74e7T^{2} \)
89 \( 1 - 6.22e3T + 6.27e7T^{2} \)
97 \( 1 + 8.16e3iT - 8.85e7T^{2} \)
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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

−12.06554372126500796298612622675, −11.12197401114257968801827912194, −10.46448661450904611303950762343, −9.190002940620614647470040760161, −8.212328645527951697811046242707, −6.95487306582569065375594637840, −6.02754932002377730313812189776, −5.02643387279504826403309031925, −3.11144982472553782786099141811, −2.23342677830943495716861635033, 0.68806548267618626938154110768, 1.75965801613710644535858023289, 3.40533919786563184003456830218, 4.74449437604296005826099456155, 6.46898995415288959396588317324, 7.06258243058874859808291019889, 8.207535223457157700234036539578, 9.383847292386609107988028225083, 10.80106124003591733599953298675, 11.13405813902959594747544408743

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