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Properties

Label 2-201-67.66-c4-0-3
Degree $2$
Conductor $201$
Sign $0.921 + 0.388i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.56i·2-s − 5.19i·3-s − 41.3·4-s + 14.1i·5-s − 39.3·6-s − 91.6i·7-s + 191. i·8-s − 27·9-s + 107.·10-s + 227. i·11-s + 214. i·12-s + 178. i·13-s − 693.·14-s + 73.7·15-s + 788.·16-s + 138.·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 0.577i·3-s − 2.58·4-s + 0.567i·5-s − 1.09·6-s − 1.87i·7-s + 2.99i·8-s − 0.333·9-s + 1.07·10-s + 1.88i·11-s + 1.49i·12-s + 1.05i·13-s − 3.53·14-s + 0.327·15-s + 3.08·16-s + 0.479·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ 0.921 + 0.388i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5010998485\)
\(L(\frac12)\) \(\approx\) \(0.5010998485\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (1.74e3 - 4.13e3i)T \)
good2 \( 1 + 7.56iT - 16T^{2} \)
5 \( 1 - 14.1iT - 625T^{2} \)
7 \( 1 + 91.6iT - 2.40e3T^{2} \)
11 \( 1 - 227. iT - 1.46e4T^{2} \)
13 \( 1 - 178. iT - 2.85e4T^{2} \)
17 \( 1 - 138.T + 8.35e4T^{2} \)
19 \( 1 + 511.T + 1.30e5T^{2} \)
23 \( 1 + 294.T + 2.79e5T^{2} \)
29 \( 1 + 46.2T + 7.07e5T^{2} \)
31 \( 1 + 900. iT - 9.23e5T^{2} \)
37 \( 1 + 1.23e3T + 1.87e6T^{2} \)
41 \( 1 - 183. iT - 2.82e6T^{2} \)
43 \( 1 - 115. iT - 3.41e6T^{2} \)
47 \( 1 - 2.10e3T + 4.87e6T^{2} \)
53 \( 1 - 2.74e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.27e3T + 1.21e7T^{2} \)
61 \( 1 - 3.17e3iT - 1.38e7T^{2} \)
71 \( 1 + 8.20e3T + 2.54e7T^{2} \)
73 \( 1 - 5.72e3T + 2.83e7T^{2} \)
79 \( 1 - 5.89e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.04e3T + 4.74e7T^{2} \)
89 \( 1 - 9.27e3T + 6.27e7T^{2} \)
97 \( 1 - 7.11e3iT - 8.85e7T^{2} \)
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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

−11.71813461373921985501348556963, −10.60225842906090170363569168610, −10.27150221395467679094609998620, −9.210866022858055906912559779302, −7.69652439943722698645555858672, −6.78649919070216765384527325395, −4.51974261475570734353390506943, −3.94099643381730379336809890309, −2.32731948806672485236664834542, −1.33174233189254019301510132912, 0.19190398909467531460894317704, 3.31237818910992027237364198835, 4.97345671005728931111266481118, 5.66753389171632719100143566329, 6.26369471891837217986342847030, 8.163356379145385703315568389929, 8.593594729329816277897600341007, 9.152297776271573838927168368515, 10.59351594848660264550669205487, 12.19523692100220754686674980364

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