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Properties

Label 2-201-67.38-c2-0-7
Degree $2$
Conductor $201$
Sign $-0.691 - 0.722i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.08 + 1.20i)2-s + 1.73i·3-s + (0.905 + 1.56i)4-s + 5.75i·5-s + (−2.08 + 3.61i)6-s + (−7.36 + 4.25i)7-s − 5.27i·8-s − 2.99·9-s + (−6.93 + 12.0i)10-s + (−5.44 + 3.14i)11-s + (−2.71 + 1.56i)12-s + (20.4 + 11.8i)13-s − 20.5·14-s − 9.97·15-s + (9.98 − 17.2i)16-s + (−4.16 + 7.21i)17-s + ⋯
L(s)  = 1  + (1.04 + 0.602i)2-s + 0.577i·3-s + (0.226 + 0.391i)4-s + 1.15i·5-s + (−0.347 + 0.602i)6-s + (−1.05 + 0.607i)7-s − 0.659i·8-s − 0.333·9-s + (−0.693 + 1.20i)10-s + (−0.495 + 0.286i)11-s + (−0.226 + 0.130i)12-s + (1.57 + 0.907i)13-s − 1.46·14-s − 0.664·15-s + (0.623 − 1.08i)16-s + (−0.245 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.691 - 0.722i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.691 - 0.722i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.852867 + 1.99590i\)
\(L(\frac12)\) \(\approx\) \(0.852867 + 1.99590i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
67 \( 1 + (2.18 + 66.9i)T \)
good2 \( 1 + (-2.08 - 1.20i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 5.75iT - 25T^{2} \)
7 \( 1 + (7.36 - 4.25i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (5.44 - 3.14i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-20.4 - 11.8i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (4.16 - 7.21i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-7.70 + 13.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.929 + 1.61i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-21.2 - 36.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-28.0 + 16.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (18.3 - 31.6i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 38.4iT - 1.84e3T^{2} \)
47 \( 1 + (7.77 + 13.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 72.6iT - 2.80e3T^{2} \)
59 \( 1 - 66.3T + 3.48e3T^{2} \)
61 \( 1 + (-26.5 - 15.3i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (-53.9 - 93.4i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-45.0 + 78.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (66.2 - 38.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (0.274 - 0.475i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 45.6T + 7.92e3T^{2} \)
97 \( 1 + (65.4 + 37.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.91245796499353732810031872283, −11.68545873540550183174814717014, −10.62102530387233214758257055992, −9.760486718927535099163096385305, −8.634555420323800112970249618356, −6.77044498199108782270535092657, −6.45742962054590052154582734573, −5.24520921245866112188458437673, −3.85667570873585651542309451312, −2.94951941062020560350787764150, 0.956261127582760694014586267516, 2.95870775806120765500078172810, 4.02911225550999163952983591684, 5.36387959256285963170579780681, 6.27467268194553032683249845913, 7.935744132485993898945070921530, 8.700403005051659581333030709581, 10.13441045440313338223495491626, 11.20859789653531276132893143363, 12.26267014133306818146195410957

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