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Properties

Label 2-201-67.30-c2-0-17
Degree $2$
Conductor $201$
Sign $-0.791 + 0.610i$
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  + (0.535 − 0.308i)2-s + 1.73i·3-s + (−1.80 + 3.13i)4-s − 6.48i·5-s + (0.535 + 0.926i)6-s + (−10.3 − 5.97i)7-s + 4.70i·8-s − 2.99·9-s + (−2.00 − 3.47i)10-s + (−0.970 − 0.560i)11-s + (−5.42 − 3.13i)12-s + (−4.61 + 2.66i)13-s − 7.38·14-s + 11.2·15-s + (−5.78 − 10.0i)16-s + (−3.61 − 6.26i)17-s + ⋯
L(s)  = 1  + (0.267 − 0.154i)2-s + 0.577i·3-s + (−0.452 + 0.783i)4-s − 1.29i·5-s + (0.0891 + 0.154i)6-s + (−1.47 − 0.853i)7-s + 0.588i·8-s − 0.333·9-s + (−0.200 − 0.347i)10-s + (−0.0882 − 0.0509i)11-s + (−0.452 − 0.261i)12-s + (−0.355 + 0.205i)13-s − 0.527·14-s + 0.749·15-s + (−0.361 − 0.626i)16-s + (−0.212 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.791 + 0.610i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.791 + 0.610i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.130098 - 0.381837i\)
\(L(\frac12)\) \(\approx\) \(0.130098 - 0.381837i\)
\(L(2)\) 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 \( 1 + (-12.2 + 65.8i)T \)
good2 \( 1 + (-0.535 + 0.308i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 6.48iT - 25T^{2} \)
7 \( 1 + (10.3 + 5.97i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (0.970 + 0.560i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (4.61 - 2.66i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (3.61 + 6.26i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (9.30 + 16.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (4.24 + 7.34i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (11.0 - 19.1i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.60 - 5.54i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (6.96 + 12.0i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-9.84 - 5.68i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 - 60.3iT - 1.84e3T^{2} \)
47 \( 1 + (35.2 - 61.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 50.0iT - 2.80e3T^{2} \)
59 \( 1 + 51.2T + 3.48e3T^{2} \)
61 \( 1 + (-78.3 + 45.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-2.42 + 4.20i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (64.5 + 111. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (88.7 + 51.2i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-71.4 - 123. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 122.T + 7.92e3T^{2} \)
97 \( 1 + (71.3 - 41.1i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.13190294488181284293871400706, −10.87670642480300463363627057369, −9.548585886603786828770318517208, −9.116978004414918847273589328619, −7.955483127204337543029558610608, −6.62185713713876748774180499075, −4.98054304110959102367628948921, −4.22387914576297260734584564744, −3.06043167888688104818483281494, −0.20062665972680223807448035124, 2.37417162077437878897855626526, 3.67708218607445529983019198965, 5.67167794854306577069528470457, 6.30503718819765395098253698794, 7.14132064037296553549460298143, 8.700518758856942033030419297120, 9.878443727345184902973109024388, 10.40894397307298269796257057415, 11.77557129800533520893934167273, 12.79689003045104433324319570994

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