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Properties

Label 2-201-67.66-c4-0-14
Degree $2$
Conductor $201$
Sign $-0.811 - 0.584i$
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  + 0.960i·2-s + 5.19i·3-s + 15.0·4-s + 34.5i·5-s − 4.99·6-s − 54.1i·7-s + 29.8i·8-s − 27·9-s − 33.2·10-s + 144. i·11-s + 78.3i·12-s + 136. i·13-s + 52.0·14-s − 179.·15-s + 212.·16-s − 424.·17-s + ⋯
L(s)  = 1  + 0.240i·2-s + 0.577i·3-s + 0.942·4-s + 1.38i·5-s − 0.138·6-s − 1.10i·7-s + 0.466i·8-s − 0.333·9-s − 0.332·10-s + 1.19i·11-s + 0.544i·12-s + 0.807i·13-s + 0.265·14-s − 0.798·15-s + 0.830·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.811 - 0.584i$
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: \((2,\ 201,\ (\ :2),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.012549501\)
\(L(\frac12)\) \(\approx\) \(2.012549501\)
\(L(3)\) 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 \( 1 + (2.62e3 - 3.64e3i)T \)
good2 \( 1 - 0.960iT - 16T^{2} \)
5 \( 1 - 34.5iT - 625T^{2} \)
7 \( 1 + 54.1iT - 2.40e3T^{2} \)
11 \( 1 - 144. iT - 1.46e4T^{2} \)
13 \( 1 - 136. iT - 2.85e4T^{2} \)
17 \( 1 + 424.T + 8.35e4T^{2} \)
19 \( 1 - 661.T + 1.30e5T^{2} \)
23 \( 1 + 351.T + 2.79e5T^{2} \)
29 \( 1 + 561.T + 7.07e5T^{2} \)
31 \( 1 + 942. iT - 9.23e5T^{2} \)
37 \( 1 + 840.T + 1.87e6T^{2} \)
41 \( 1 - 2.06e3iT - 2.82e6T^{2} \)
43 \( 1 - 3.10e3iT - 3.41e6T^{2} \)
47 \( 1 + 963.T + 4.87e6T^{2} \)
53 \( 1 + 2.78e3iT - 7.89e6T^{2} \)
59 \( 1 + 234.T + 1.21e7T^{2} \)
61 \( 1 + 2.34e3iT - 1.38e7T^{2} \)
71 \( 1 - 1.79e3T + 2.54e7T^{2} \)
73 \( 1 + 6.59e3T + 2.83e7T^{2} \)
79 \( 1 + 4.55e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.16e3T + 4.74e7T^{2} \)
89 \( 1 - 1.32e4T + 6.27e7T^{2} \)
97 \( 1 - 1.31e4iT - 8.85e7T^{2} \)
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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

−11.64154174993570891075547446485, −11.23116666600902646939077984227, −10.26347990738406501666469585012, −9.587052782191579145277669503249, −7.66598254793295808261034865555, −7.07245341685735423890625837938, −6.27722472385595792643586255284, −4.55631895314306527057170448153, −3.30775135801531107370535335842, −2.00645563544195939209758700170, 0.67000423278320704838813591381, 1.95860117147999020071023548943, 3.27432571670403063576212328683, 5.31101203348044035938800370109, 5.93675348690981775175259300684, 7.35903927670723157124533586179, 8.511345135130955324806101354763, 9.096560943198415238959781971482, 10.63228419588157508213345553064, 11.75621783328898516166604499642

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