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Properties

Label 2-201-67.30-c2-0-15
Degree $2$
Conductor $201$
Sign $0.907 + 0.421i$
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  + (3.30 − 1.91i)2-s + 1.73i·3-s + (5.30 − 9.18i)4-s + 8.30i·5-s + (3.30 + 5.73i)6-s + (8.95 + 5.17i)7-s − 25.2i·8-s − 2.99·9-s + (15.8 + 27.4i)10-s + (−14.8 − 8.57i)11-s + (15.9 + 9.18i)12-s + (1.88 − 1.08i)13-s + 39.5·14-s − 14.3·15-s + (−27.0 − 46.7i)16-s + (−10.0 − 17.4i)17-s + ⋯
L(s)  = 1  + (1.65 − 0.955i)2-s + 0.577i·3-s + (1.32 − 2.29i)4-s + 1.66i·5-s + (0.551 + 0.955i)6-s + (1.27 + 0.738i)7-s − 3.15i·8-s − 0.333·9-s + (1.58 + 2.74i)10-s + (−1.35 − 0.779i)11-s + (1.32 + 0.765i)12-s + (0.144 − 0.0835i)13-s + 2.82·14-s − 0.959·15-s + (−1.68 − 2.92i)16-s + (−0.592 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.421i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.907 + 0.421i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.907 + 0.421i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.72516 - 0.822622i\)
\(L(\frac12)\) \(\approx\) \(3.72516 - 0.822622i\)
\(L(2)\) 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 \( 1 + (64.0 - 19.5i)T \)
good2 \( 1 + (-3.30 + 1.91i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 8.30iT - 25T^{2} \)
7 \( 1 + (-8.95 - 5.17i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.8 + 8.57i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.88 + 1.08i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (10.0 + 17.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.53 - 2.65i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (5.78 + 10.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.3 + 28.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (6.11 + 3.52i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.3 - 43.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (4.41 + 2.54i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 - 57.4iT - 1.84e3T^{2} \)
47 \( 1 + (13.6 - 23.5i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 2.48iT - 2.80e3T^{2} \)
59 \( 1 - 40.6T + 3.48e3T^{2} \)
61 \( 1 + (3.67 - 2.12i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-46.5 + 80.6i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-25.2 - 43.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (60.8 + 35.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-63.1 - 109. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 87.5T + 7.92e3T^{2} \)
97 \( 1 + (-89.0 + 51.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.82880589265339955409948662383, −11.24439254480754410719264311626, −10.77001209346667252172524677614, −9.871101903670834194646572317224, −7.969833382491282371647888069549, −6.41649523244068982688348915139, −5.47210370407787946572814861394, −4.51189347070044104510157188687, −2.99715998298822042603155220676, −2.44422256941951068764649487834, 1.91771020643188157382987916293, 4.11839769814609439438476522705, 4.91964016488807514569439864320, 5.58854607764754170528893784336, 7.15099188586971451607637006873, 7.940349084018687312869187043601, 8.583478649312904644299264395214, 10.80006468941799694340188854708, 11.93263238234197424099249573994, 12.72867052175245661473842273345

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