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Properties

Label 2-201-201.35-c2-0-4
Degree $2$
Conductor $201$
Sign $-0.839 - 0.542i$
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  + (0.426 + 2.96i)3-s + (−3.55 − 1.83i)4-s + (6.63 + 6.33i)7-s + (−8.63 + 2.53i)9-s + (3.92 − 11.3i)12-s + (−25.0 + 4.82i)13-s + (9.28 + 13.0i)16-s + (−22.5 + 21.4i)19-s + (−15.9 + 22.4i)21-s + (−16.3 − 18.8i)25-s + (−11.2 − 24.5i)27-s + (−12.0 − 34.6i)28-s + (48.8 + 9.41i)31-s + (35.3 + 6.81i)36-s + (−17.6 + 30.5i)37-s + ⋯
L(s)  = 1  + (0.142 + 0.989i)3-s + (−0.888 − 0.458i)4-s + (0.948 + 0.904i)7-s + (−0.959 + 0.281i)9-s + (0.327 − 0.945i)12-s + (−1.92 + 0.370i)13-s + (0.580 + 0.814i)16-s + (−1.18 + 1.13i)19-s + (−0.760 + 1.06i)21-s + (−0.654 − 0.755i)25-s + (−0.415 − 0.909i)27-s + (−0.428 − 1.23i)28-s + (1.57 + 0.303i)31-s + (0.981 + 0.189i)36-s + (−0.477 + 0.827i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.839 - 0.542i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.839 - 0.542i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.239005 + 0.809781i\)
\(L(\frac12)\) \(\approx\) \(0.239005 + 0.809781i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.426 - 2.96i)T \)
67 \( 1 + (-6.23 - 66.7i)T \)
good2 \( 1 + (3.55 + 1.83i)T^{2} \)
5 \( 1 + (16.3 + 18.8i)T^{2} \)
7 \( 1 + (-6.63 - 6.33i)T + (2.33 + 48.9i)T^{2} \)
11 \( 1 + (-118. + 22.8i)T^{2} \)
13 \( 1 + (25.0 - 4.82i)T + (156. - 62.8i)T^{2} \)
17 \( 1 + (-167. + 235. i)T^{2} \)
19 \( 1 + (22.5 - 21.4i)T + (17.1 - 360. i)T^{2} \)
23 \( 1 + (-382. - 365. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-48.8 - 9.41i)T + (892. + 357. i)T^{2} \)
37 \( 1 + (17.6 - 30.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (1.67e3 + 159. i)T^{2} \)
43 \( 1 + (-72.1 - 46.3i)T + (768. + 1.68e3i)T^{2} \)
47 \( 1 + (-520. + 2.14e3i)T^{2} \)
53 \( 1 + (-1.16e3 + 2.55e3i)T^{2} \)
59 \( 1 + (495. + 3.44e3i)T^{2} \)
61 \( 1 + (-117. - 11.2i)T + (3.65e3 + 704. i)T^{2} \)
71 \( 1 + (-2.92e3 - 4.10e3i)T^{2} \)
73 \( 1 + (85.7 + 8.19i)T + (5.23e3 + 1.00e3i)T^{2} \)
79 \( 1 + (-49.8 + 144. i)T + (-4.90e3 - 3.85e3i)T^{2} \)
83 \( 1 + (2.25e3 - 6.51e3i)T^{2} \)
89 \( 1 + (7.60e3 + 2.23e3i)T^{2} \)
97 \( 1 + (36.9 - 64.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.43813921818181488891722446055, −11.66743110097295390132566921765, −10.32660131834131025991133669563, −9.776686668757944934626436260399, −8.718740613725668461157144413125, −8.025343669685443690840582538694, −5.99151464689676763931185894097, −4.94065427339216404957758798148, −4.30720776213274721707771488759, −2.34436278224257291676683677293, 0.47030766639054917410723971006, 2.45735362913757941066270957210, 4.23432813557673820615227844834, 5.27282508410653439080751381709, 7.06416436090082764399516332076, 7.69595701098986620614813923015, 8.577853928468303798566579537596, 9.723210112031983624339841032319, 10.98847118941850826197563370778, 12.08849496499023820621108731199

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