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Properties

Label 2-201-201.200-c3-0-63
Degree $2$
Conductor $201$
Sign $-0.596 - 0.802i$
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  − 5.19i·3-s − 8·4-s − 31.1i·7-s − 27·9-s + 41.5i·12-s + 62.3i·13-s + 64·16-s − 56·19-s − 162·21-s − 125·25-s + 140. i·27-s + 249. i·28-s + 155. i·31-s + 216·36-s − 110·37-s + ⋯
L(s)  = 1  − 0.999i·3-s − 4-s − 1.68i·7-s − 9-s + 0.999i·12-s + 1.33i·13-s + 16-s − 0.676·19-s − 1.68·21-s − 25-s + 1.00i·27-s + 1.68i·28-s + 0.903i·31-s + 36-s − 0.488·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.596 - 0.802i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.596 - 0.802i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0996288 + 0.198301i\)
\(L(\frac12)\) \(\approx\) \(0.0996288 + 0.198301i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (-440 + 327. i)T \)
good2 \( 1 + 8T^{2} \)
5 \( 1 + 125T^{2} \)
7 \( 1 + 31.1iT - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 62.3iT - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 + 56T + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + 110T + 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 218. iT - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 + 935. iT - 2.26e5T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + 1.19e3T + 3.89e5T^{2} \)
79 \( 1 + 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
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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

−11.44301739480388808777812115562, −10.38549000797420859598069883556, −9.281383711977706976661445029648, −8.209362677422026480680892802523, −7.28188507526970605936651403810, −6.33780361926974888504677185224, −4.72165031413561355605528058130, −3.67670025282945162745938213002, −1.51240198732345375275600992910, −0.10053276823414201082055138696, 2.70075135303811268671558316027, 4.02595647914943665886086498940, 5.34860825482106459201274946761, 5.81170062061517169817135730251, 8.113291644496579265846204380921, 8.756596408526256011034653092661, 9.601211249118831312829437356598, 10.44031076525498834467129316778, 11.70197924022879915118676770121, 12.57965691451120960661235171515

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