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Properties

Label 2-201-201.137-c3-0-27
Degree $2$
Conductor $201$
Sign $0.856 - 0.515i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.14 + 0.739i)3-s + (−6.73 − 4.32i)4-s + (−4.34 + 14.7i)7-s + (25.9 + 7.60i)9-s + (−31.4 − 27.2i)12-s + (60.1 + 52.0i)13-s + (26.5 + 58.2i)16-s + (140. − 41.2i)19-s + (−33.2 + 72.8i)21-s + (81.8 − 94.4i)25-s + (127. + 58.2i)27-s + (93.1 − 80.7i)28-s + (−162. + 141. i)31-s + (−141. − 163. i)36-s − 257.·37-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)3-s + (−0.841 − 0.540i)4-s + (−0.234 + 0.798i)7-s + (0.959 + 0.281i)9-s + (−0.755 − 0.654i)12-s + (1.28 + 1.11i)13-s + (0.415 + 0.909i)16-s + (1.69 − 0.497i)19-s + (−0.345 + 0.757i)21-s + (0.654 − 0.755i)25-s + (0.909 + 0.415i)27-s + (0.629 − 0.545i)28-s + (−0.943 + 0.817i)31-s + (−0.654 − 0.755i)36-s − 1.14·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.03024 + 0.563476i\)
\(L(\frac12)\) \(\approx\) \(2.03024 + 0.563476i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.14 - 0.739i)T \)
67 \( 1 + (40.7 - 546. i)T \)
good2 \( 1 + (6.73 + 4.32i)T^{2} \)
5 \( 1 + (-81.8 + 94.4i)T^{2} \)
7 \( 1 + (4.34 - 14.7i)T + (-288. - 185. i)T^{2} \)
11 \( 1 + (-871. + 1.00e3i)T^{2} \)
13 \( 1 + (-60.1 - 52.0i)T + (312. + 2.17e3i)T^{2} \)
17 \( 1 + (-2.04e3 + 4.46e3i)T^{2} \)
19 \( 1 + (-140. + 41.2i)T + (5.77e3 - 3.70e3i)T^{2} \)
23 \( 1 + (1.16e4 + 3.42e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (162. - 141. i)T + (4.23e3 - 2.94e4i)T^{2} \)
37 \( 1 + 257.T + 5.06e4T^{2} \)
41 \( 1 + (2.86e4 - 6.26e4i)T^{2} \)
43 \( 1 + (-135. - 210. i)T + (-3.30e4 + 7.23e4i)T^{2} \)
47 \( 1 + (9.96e4 + 2.92e4i)T^{2} \)
53 \( 1 + (6.18e4 + 1.35e5i)T^{2} \)
59 \( 1 + (2.92e4 - 2.03e5i)T^{2} \)
61 \( 1 + (42.7 + 19.5i)T + (1.48e5 + 1.71e5i)T^{2} \)
71 \( 1 + (-1.48e5 - 3.25e5i)T^{2} \)
73 \( 1 + (63.9 - 140. i)T + (-2.54e5 - 2.93e5i)T^{2} \)
79 \( 1 + (1.04e3 + 905. i)T + (7.01e4 + 4.88e5i)T^{2} \)
83 \( 1 + (3.74e5 - 4.32e5i)T^{2} \)
89 \( 1 + (6.76e5 - 1.98e5i)T^{2} \)
97 \( 1 + 1.12e3iT - 9.12e5T^{2} \)
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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

−12.25781185470456354958445273028, −10.95430094561503443893617367194, −9.781390844744064981275543298778, −9.010306418141829660243608182640, −8.529995914671086586767527000313, −7.00602656478795960937527964146, −5.64840843480868933304339338621, −4.41535885536488132060868514981, −3.18420573304324750408282232039, −1.46121951314796661695581790896, 1.00607380950368537451596985180, 3.26370952538141776013736957696, 3.83239632426898088750754711173, 5.41960731626013560866239677073, 7.19392586264716216538772450104, 7.921170295365932050885873571731, 8.860016183937050390612822326984, 9.733381786071717723119710500572, 10.73831840382944767544315970996, 12.23996137639079778989665832795

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