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Properties

Label 2-201-67.29-c3-0-33
Degree $2$
Conductor $201$
Sign $0.148 - 0.988i$
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  + (2.68 − 4.65i)2-s − 3·3-s + (−10.4 − 18.1i)4-s − 6.42·5-s + (−8.06 + 13.9i)6-s + (−7.03 − 12.1i)7-s − 69.4·8-s + 9·9-s + (−17.2 + 29.9i)10-s + (20.0 + 34.6i)11-s + (31.3 + 54.3i)12-s + (33.8 − 58.6i)13-s − 75.6·14-s + 19.2·15-s + (−102. + 178. i)16-s + (−54.4 + 94.3i)17-s + ⋯
L(s)  = 1  + (0.950 − 1.64i)2-s − 0.577·3-s + (−1.30 − 2.26i)4-s − 0.574·5-s + (−0.548 + 0.950i)6-s + (−0.379 − 0.657i)7-s − 3.06·8-s + 0.333·9-s + (−0.546 + 0.946i)10-s + (0.548 + 0.950i)11-s + (0.754 + 1.30i)12-s + (0.722 − 1.25i)13-s − 1.44·14-s + 0.331·15-s + (−1.60 + 2.78i)16-s + (−0.776 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.148 - 0.988i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.604093 + 0.520253i\)
\(L(\frac12)\) \(\approx\) \(0.604093 + 0.520253i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
67 \( 1 + (252. + 486. i)T \)
good2 \( 1 + (-2.68 + 4.65i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 6.42T + 125T^{2} \)
7 \( 1 + (7.03 + 12.1i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-20.0 - 34.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-33.8 + 58.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (54.4 - 94.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (22.1 - 38.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-38.2 + 66.2i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (23.4 + 40.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (81.6 + 141. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (120. - 208. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (182. + 316. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 402.T + 7.95e4T^{2} \)
47 \( 1 + (142. + 247. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 433.T + 1.48e5T^{2} \)
59 \( 1 - 518.T + 2.05e5T^{2} \)
61 \( 1 + (-60.7 + 105. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (103. + 179. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (389. - 673. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (169. + 293. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (476. - 825. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 254.T + 7.04e5T^{2} \)
97 \( 1 + (315. - 545. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.37889447916609351612875946755, −10.43584228178086556896720323635, −10.04127255258813370450947283869, −8.483352921981476118103525413992, −6.69038371893878673993506892510, −5.52165476670403977703838043788, −4.19394859861891230012732966507, −3.62718069255202664902778506944, −1.77319890271515986442190213943, −0.27160105585120944776535250193, 3.39868158505871933428270011088, 4.49307812468702135771639803034, 5.58459774012585025331544384704, 6.53887775250898929296279462244, 7.17603567637799617752805467437, 8.621670672245189803983314474846, 9.173473388685121350174487776827, 11.46731130814672012444875802348, 11.79766462467665062419669804379, 13.10753817581253282087887589580

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