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Properties

Label 2-201-67.29-c3-0-21
Degree $2$
Conductor $201$
Sign $0.582 - 0.813i$
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  + (−2.00 + 3.47i)2-s + 3·3-s + (−4.04 − 7.00i)4-s + 19.8·5-s + (−6.01 + 10.4i)6-s + (−14.1 − 24.5i)7-s + 0.378·8-s + 9·9-s + (−39.9 + 69.1i)10-s + (24.4 + 42.2i)11-s + (−12.1 − 21.0i)12-s + (32.8 − 56.9i)13-s + 113.·14-s + 59.6·15-s + (31.6 − 54.7i)16-s + (16.2 − 28.2i)17-s + ⋯
L(s)  = 1  + (−0.709 + 1.22i)2-s + 0.577·3-s + (−0.505 − 0.876i)4-s + 1.77·5-s + (−0.409 + 0.709i)6-s + (−0.764 − 1.32i)7-s + 0.0167·8-s + 0.333·9-s + (−1.26 + 2.18i)10-s + (0.669 + 1.15i)11-s + (−0.292 − 0.505i)12-s + (0.701 − 1.21i)13-s + 2.16·14-s + 1.02·15-s + (0.494 − 0.855i)16-s + (0.232 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.582 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.582 - 0.813i$
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: \((2,\ 201,\ (\ :3/2),\ 0.582 - 0.813i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69101 + 0.868908i\)
\(L(\frac12)\) \(\approx\) \(1.69101 + 0.868908i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.05 - 548. i)T \)
good2 \( 1 + (2.00 - 3.47i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 19.8T + 125T^{2} \)
7 \( 1 + (14.1 + 24.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-24.4 - 42.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-32.8 + 56.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-16.2 + 28.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.0 - 20.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-73.9 + 128. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-9.06 - 15.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-97.8 - 169. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (148. - 257. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-159. - 277. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 192.T + 7.95e4T^{2} \)
47 \( 1 + (177. + 306. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 316.T + 1.48e5T^{2} \)
59 \( 1 + 526.T + 2.05e5T^{2} \)
61 \( 1 + (-432. + 749. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (-342. - 593. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-54.0 + 93.5i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (365. + 632. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (164. - 284. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.52e3T + 7.04e5T^{2} \)
97 \( 1 + (624. - 1.08e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.62913567545604946644915143699, −10.34816442860536928240542171071, −9.982273218173831564591585938788, −9.164454835972865117773145007253, −8.088659596613941703154021201964, −6.75898148617503552488190543876, −6.51461443533435131855865866386, −5.01663157220584718058714156305, −3.09135349520568799482701873464, −1.15909806337246455853995729622, 1.45794558236542501715803224015, 2.38830557032547235383724215509, 3.43820519887817893728997954478, 5.79242963252554544244145743394, 6.36714038718096350331174361614, 8.666606189173430178479430166364, 9.239879216805420925840624687469, 9.521946749708913689235493870606, 10.75820424828932464807324740535, 11.69250469196953279593971632225

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