Properties

Label 2-201-67.30-c2-0-11
Degree $2$
Conductor $201$
Sign $0.966 - 0.257i$
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  + (2.00 − 1.15i)2-s + 1.73i·3-s + (0.672 − 1.16i)4-s − 1.12i·5-s + (2.00 + 3.46i)6-s + (6.42 + 3.71i)7-s + 6.13i·8-s − 2.99·9-s + (−1.30 − 2.25i)10-s + (11.9 + 6.90i)11-s + (2.01 + 1.16i)12-s + (4.14 − 2.39i)13-s + 17.1·14-s + 1.94·15-s + (9.78 + 16.9i)16-s + (−14.7 − 25.5i)17-s + ⋯
L(s)  = 1  + (1.00 − 0.577i)2-s + 0.577i·3-s + (0.168 − 0.291i)4-s − 0.225i·5-s + (0.333 + 0.577i)6-s + (0.918 + 0.530i)7-s + 0.767i·8-s − 0.333·9-s + (−0.130 − 0.225i)10-s + (1.08 + 0.627i)11-s + (0.168 + 0.0970i)12-s + (0.318 − 0.183i)13-s + 1.22·14-s + 0.129·15-s + (0.611 + 1.05i)16-s + (−0.867 − 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.966 - 0.257i$
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: \((2,\ 201,\ (\ :1),\ 0.966 - 0.257i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.63849 + 0.345846i\)
\(L(\frac12)\) \(\approx\) \(2.63849 + 0.345846i\)
\(L(2)\) 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 \( 1 + (36.6 - 56.0i)T \)
good2 \( 1 + (-2.00 + 1.15i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 1.12iT - 25T^{2} \)
7 \( 1 + (-6.42 - 3.71i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-11.9 - 6.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.14 + 2.39i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (14.7 + 25.5i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.44 + 2.50i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (1.08 + 1.87i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (21.0 - 36.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-1.73 - 1.00i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (30.4 + 52.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (59.5 + 34.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 - 38.8iT - 1.84e3T^{2} \)
47 \( 1 + (-27.0 + 46.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 73.4iT - 2.80e3T^{2} \)
59 \( 1 + 37.8T + 3.48e3T^{2} \)
61 \( 1 + (-3.26 + 1.88i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-37.1 + 64.2i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-57.4 - 99.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-39.4 - 22.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (46.1 + 79.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + (-10.6 + 6.16i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.12535800085633223299132582205, −11.52292600882062737596161693615, −10.72032245191459163944765814093, −9.177307136940819016284971365699, −8.586443462822737332733002952643, −6.98297324772258050278082731318, −5.29958409006430904708518839512, −4.72839233794037337254376679206, −3.55354285944326491571945993510, −2.04823535903789525875371901911, 1.41669018483825172146330091464, 3.65416958121672095014917803941, 4.64734459472868002168238174146, 6.09256725240530223726972265816, 6.65615613555004413262844248239, 7.892750738471277550907512237880, 8.940655048412609723296629091207, 10.48656342503981945825109290982, 11.40542338527735544379629129501, 12.39890664059656217450367693064

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