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Properties

Label 2-201-67.38-c2-0-5
Degree $2$
Conductor $201$
Sign $-0.592 - 0.805i$
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  + (−0.443 − 0.255i)2-s + 1.73i·3-s + (−1.86 − 3.23i)4-s + 5.57i·5-s + (0.443 − 0.767i)6-s + (3.11 − 1.79i)7-s + 3.95i·8-s − 2.99·9-s + (1.42 − 2.47i)10-s + (−8.16 + 4.71i)11-s + (5.60 − 3.23i)12-s + (−3.89 − 2.25i)13-s − 1.84·14-s − 9.65·15-s + (−6.46 + 11.1i)16-s + (−12.0 + 20.8i)17-s + ⋯
L(s)  = 1  + (−0.221 − 0.127i)2-s + 0.577i·3-s + (−0.467 − 0.809i)4-s + 1.11i·5-s + (0.0738 − 0.127i)6-s + (0.445 − 0.257i)7-s + 0.494i·8-s − 0.333·9-s + (0.142 − 0.247i)10-s + (−0.741 + 0.428i)11-s + (0.467 − 0.269i)12-s + (−0.299 − 0.173i)13-s − 0.131·14-s − 0.643·15-s + (−0.403 + 0.699i)16-s + (−0.709 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.328376 + 0.649464i\)
\(L(\frac12)\) \(\approx\) \(0.328376 + 0.649464i\)
\(L(2)\) 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 \( 1 + (-6.42 + 66.6i)T \)
good2 \( 1 + (0.443 + 0.255i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 5.57iT - 25T^{2} \)
7 \( 1 + (-3.11 + 1.79i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.16 - 4.71i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (3.89 + 2.25i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (12.0 - 20.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (6.97 - 12.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (5.76 - 9.98i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-1.52 - 2.63i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (37.2 - 21.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-0.844 + 1.46i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-36.4 + 21.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 46.0iT - 1.84e3T^{2} \)
47 \( 1 + (-22.2 - 38.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 29.2iT - 2.80e3T^{2} \)
59 \( 1 - 52.6T + 3.48e3T^{2} \)
61 \( 1 + (-27.8 - 16.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (18.2 + 31.6i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-19.0 + 32.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-57.0 + 32.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (32.6 - 56.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 45.8T + 7.92e3T^{2} \)
97 \( 1 + (-135. - 78.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.55614505757640749443212460944, −10.92200985209384016228477999510, −10.71794075319143039716479172056, −9.892885127699813513183051495220, −8.740103223902517889391684192502, −7.56883028926490238302286208865, −6.22375179437281780729311903401, −5.11754126632073702316609510249, −3.86474310161978909494521120805, −2.11257146991511659130529169122, 0.43561272223712275591919675607, 2.54949159630497548503544837444, 4.40707698949297954288048782369, 5.34113761217355143453256860655, 6.99576385189850180386528367473, 8.009278009497467029826256548985, 8.719977771805023125209071825310, 9.481675570087521167416503442419, 11.19138548123346352561405769268, 12.03255179383525717145197100608

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