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Properties

Label 2-201-67.29-c3-0-4
Degree $2$
Conductor $201$
Sign $-0.984 + 0.173i$
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  + (−0.829 + 1.43i)2-s − 3·3-s + (2.62 + 4.54i)4-s − 7.05·5-s + (2.48 − 4.31i)6-s + (18.3 + 31.8i)7-s − 21.9·8-s + 9·9-s + (5.85 − 10.1i)10-s + (11.1 + 19.3i)11-s + (−7.86 − 13.6i)12-s + (31.4 − 54.5i)13-s − 61.0·14-s + 21.1·15-s + (−2.73 + 4.72i)16-s + (−45.1 + 78.2i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.508i)2-s − 0.577·3-s + (0.327 + 0.567i)4-s − 0.630·5-s + (0.169 − 0.293i)6-s + (0.993 + 1.72i)7-s − 0.971·8-s + 0.333·9-s + (0.185 − 0.320i)10-s + (0.306 + 0.530i)11-s + (−0.189 − 0.327i)12-s + (0.671 − 1.16i)13-s − 1.16·14-s + 0.364·15-s + (−0.0426 + 0.0739i)16-s + (−0.644 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.984 + 0.173i$
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: \((2,\ 201,\ (\ :3/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0768396 - 0.877556i\)
\(L(\frac12)\) \(\approx\) \(0.0768396 - 0.877556i\)
\(L(\frac{5}{2})\) 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 \( 1 + (381. - 394. i)T \)
good2 \( 1 + (0.829 - 1.43i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 7.05T + 125T^{2} \)
7 \( 1 + (-18.3 - 31.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-11.1 - 19.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-31.4 + 54.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (45.1 - 78.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-13.0 + 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (67.0 - 116. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (124. + 216. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-59.8 - 103. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-66.0 + 114. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (190. + 330. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 209.T + 7.95e4T^{2} \)
47 \( 1 + (-158. - 274. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 393.T + 1.48e5T^{2} \)
59 \( 1 + 138.T + 2.05e5T^{2} \)
61 \( 1 + (-64.3 + 111. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (-222. - 385. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-554. + 961. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-476. - 824. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-290. + 502. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 525.T + 7.04e5T^{2} \)
97 \( 1 + (743. - 1.28e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.09500198806973634909837132318, −11.79741269024650321901862845537, −10.84539395753091990519086786604, −9.226325826293654416350766423372, −8.260353952853939865439218415977, −7.70166071197381545540499203668, −6.21381276350453939427381057555, −5.44654329633108560971337890807, −3.83250673415696810563839519885, −2.10978932142166938255629127063, 0.45472007686442450528198854403, 1.59853364354231437920280712259, 3.81301136742429465533593817000, 4.82629945589806118130310002808, 6.43250647216680851994892162313, 7.23901446707832573944935974149, 8.518561575995018251679878390913, 9.831356081780323462178148889845, 10.79473337286200290138766204556, 11.37851608968434182430529762935

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