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Properties

Label 2-201-201.116-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.382 - 0.924i$
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  + (−2.52 − 1.62i)3-s + (−1.30 − 3.78i)4-s + (−9.12 − 0.870i)7-s + (3.73 + 8.18i)9-s + (−2.82 + 11.6i)12-s + (14.8 + 14.2i)13-s + (−12.5 + 9.89i)16-s + (−6.85 + 0.654i)19-s + (21.6 + 16.9i)21-s + (−23.9 + 7.04i)25-s + (3.84 − 26.7i)27-s + (8.64 + 35.6i)28-s + (−42.0 + 40.1i)31-s + (26.0 − 24.8i)36-s + (2.71 + 4.70i)37-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)3-s + (−0.327 − 0.945i)4-s + (−1.30 − 0.124i)7-s + (0.415 + 0.909i)9-s + (−0.235 + 0.971i)12-s + (1.14 + 1.09i)13-s + (−0.786 + 0.618i)16-s + (−0.360 + 0.0344i)19-s + (1.02 + 0.809i)21-s + (−0.959 + 0.281i)25-s + (0.142 − 0.989i)27-s + (0.308 + 1.27i)28-s + (−1.35 + 1.29i)31-s + (0.723 − 0.690i)36-s + (0.0733 + 0.127i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.382 - 0.924i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.382 - 0.924i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0684473 + 0.102395i\)
\(L(\frac12)\) \(\approx\) \(0.0684473 + 0.102395i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.52 + 1.62i)T \)
67 \( 1 + (-41.3 + 52.7i)T \)
good2 \( 1 + (1.30 + 3.78i)T^{2} \)
5 \( 1 + (23.9 - 7.04i)T^{2} \)
7 \( 1 + (9.12 + 0.870i)T + (48.1 + 9.27i)T^{2} \)
11 \( 1 + (-87.5 - 83.4i)T^{2} \)
13 \( 1 + (-14.8 - 14.2i)T + (8.04 + 168. i)T^{2} \)
17 \( 1 + (227. + 178. i)T^{2} \)
19 \( 1 + (6.85 - 0.654i)T + (354. - 68.3i)T^{2} \)
23 \( 1 + (526. + 50.2i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (42.0 - 40.1i)T + (45.7 - 959. i)T^{2} \)
37 \( 1 + (-2.71 - 4.70i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-1.56e3 + 624. i)T^{2} \)
43 \( 1 + (49.5 + 57.1i)T + (-263. + 1.83e3i)T^{2} \)
47 \( 1 + (-1.28e3 + 1.79e3i)T^{2} \)
53 \( 1 + (399. + 2.78e3i)T^{2} \)
59 \( 1 + (-2.92e3 - 1.88e3i)T^{2} \)
61 \( 1 + (19.8 - 7.92i)T + (2.69e3 - 2.56e3i)T^{2} \)
71 \( 1 + (3.96e3 - 3.11e3i)T^{2} \)
73 \( 1 + (45.9 - 18.3i)T + (3.85e3 - 3.67e3i)T^{2} \)
79 \( 1 + (35.5 - 146. i)T + (-5.54e3 - 2.85e3i)T^{2} \)
83 \( 1 + (-1.62e3 + 6.69e3i)T^{2} \)
89 \( 1 + (-3.29e3 + 7.20e3i)T^{2} \)
97 \( 1 + (96.9 + 167. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.61455023893496989149037276544, −11.45882610120774495115821754734, −10.62032189349177733283348749336, −9.721419089432937048147616647970, −8.717356064248901798670409516731, −6.95949743690040353229268101184, −6.33077691694229251204231827313, −5.39734451969924368992750942670, −3.91574757626017755070557834967, −1.63141445960952292160365040494, 0.07851029381438647948902911721, 3.22301864309543482865489982966, 4.07873116431680434822075597369, 5.65142442425933128021126010021, 6.54148299692914437512397822528, 7.891081431219813810048007347873, 9.100092855400488699361172687276, 9.922037135565819719358678511918, 10.99849589233309845489180426889, 11.95474401948633372129654230517

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