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Properties

Label 2-201-67.37-c3-0-7
Degree $2$
Conductor $201$
Sign $0.359 - 0.933i$
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  + (0.824 + 1.42i)2-s − 3·3-s + (2.64 − 4.57i)4-s − 20.6·5-s + (−2.47 − 4.28i)6-s + (1.21 − 2.10i)7-s + 21.9·8-s + 9·9-s + (−17.0 − 29.5i)10-s + (3.15 − 5.46i)11-s + (−7.92 + 13.7i)12-s + (39.6 + 68.7i)13-s + 4.00·14-s + 62.0·15-s + (−3.06 − 5.31i)16-s + (59.8 + 103. i)17-s + ⋯
L(s)  = 1  + (0.291 + 0.504i)2-s − 0.577·3-s + (0.330 − 0.571i)4-s − 1.84·5-s + (−0.168 − 0.291i)6-s + (0.0655 − 0.113i)7-s + 0.967·8-s + 0.333·9-s + (−0.539 − 0.934i)10-s + (0.0865 − 0.149i)11-s + (−0.190 + 0.330i)12-s + (0.846 + 1.46i)13-s + 0.0764·14-s + 1.06·15-s + (−0.0479 − 0.0829i)16-s + (0.854 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.359 - 0.933i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.359 - 0.933i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.05273 + 0.722724i\)
\(L(\frac12)\) \(\approx\) \(1.05273 + 0.722724i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-459. + 298. i)T \)
good2 \( 1 + (-0.824 - 1.42i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 20.6T + 125T^{2} \)
7 \( 1 + (-1.21 + 2.10i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-3.15 + 5.46i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-39.6 - 68.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-59.8 - 103. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (44.2 + 76.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-71.8 - 124. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (32.3 - 56.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-36.7 + 63.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (64.8 + 112. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (38.7 - 67.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 - 235.T + 7.95e4T^{2} \)
47 \( 1 + (245. - 424. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 431.T + 1.48e5T^{2} \)
59 \( 1 + 728.T + 2.05e5T^{2} \)
61 \( 1 + (-192. - 332. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (266. - 460. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (25.6 + 44.5i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-257. + 446. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (85.9 + 148. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.10e3T + 7.04e5T^{2} \)
97 \( 1 + (9.05 + 15.6i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.96010137741911404448022059033, −11.17603269245314795799317562404, −10.73813619655357273741753363197, −9.053663974015511815702638813968, −7.81961233074583315397817299848, −6.99690678236364840255853206112, −6.03826025270339292420250897234, −4.62144358979263994542967005961, −3.79386487304460460617814302897, −1.20941463096145193011139740190, 0.65991223023067931346096665842, 3.05763162058921617850092307599, 3.93498974048091473630644760047, 5.11596699724042408236187537418, 6.85439126973323926770479601741, 7.78680128469054048906578994410, 8.436317288414234653033268297551, 10.40418634929311329037142979085, 11.05359708236662974151726431156, 12.01828595202249177373271797536

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