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Properties

Label 2-201-3.2-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.127 + 0.991i$
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  + 2.85i·2-s + (−0.383 + 2.97i)3-s − 4.13·4-s − 2.94i·5-s + (−8.48 − 1.09i)6-s − 7.08·7-s − 0.391i·8-s + (−8.70 − 2.27i)9-s + 8.39·10-s − 11.8i·11-s + (1.58 − 12.3i)12-s − 12.0·13-s − 20.2i·14-s + (8.76 + 1.12i)15-s − 15.4·16-s + 19.5i·17-s + ⋯
L(s)  = 1  + 1.42i·2-s + (−0.127 + 0.991i)3-s − 1.03·4-s − 0.588i·5-s + (−1.41 − 0.182i)6-s − 1.01·7-s − 0.0489i·8-s + (−0.967 − 0.253i)9-s + 0.839·10-s − 1.08i·11-s + (0.132 − 1.02i)12-s − 0.924·13-s − 1.44i·14-s + (0.584 + 0.0751i)15-s − 0.964·16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.127 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.353854 - 0.402326i\)
\(L(\frac12)\) \(\approx\) \(0.353854 - 0.402326i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.383 - 2.97i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 - 2.85iT - 4T^{2} \)
5 \( 1 + 2.94iT - 25T^{2} \)
7 \( 1 + 7.08T + 49T^{2} \)
11 \( 1 + 11.8iT - 121T^{2} \)
13 \( 1 + 12.0T + 169T^{2} \)
17 \( 1 - 19.5iT - 289T^{2} \)
19 \( 1 + 10.1T + 361T^{2} \)
23 \( 1 - 45.8iT - 529T^{2} \)
29 \( 1 - 2.17iT - 841T^{2} \)
31 \( 1 - 39.1T + 961T^{2} \)
37 \( 1 + 50.6T + 1.36e3T^{2} \)
41 \( 1 - 27.7iT - 1.68e3T^{2} \)
43 \( 1 + 21.8T + 1.84e3T^{2} \)
47 \( 1 - 34.9iT - 2.20e3T^{2} \)
53 \( 1 - 44.7iT - 2.80e3T^{2} \)
59 \( 1 + 56.8iT - 3.48e3T^{2} \)
61 \( 1 + 31.9T + 3.72e3T^{2} \)
71 \( 1 + 84.4iT - 5.04e3T^{2} \)
73 \( 1 + 110.T + 5.32e3T^{2} \)
79 \( 1 - 118.T + 6.24e3T^{2} \)
83 \( 1 - 59.6iT - 6.88e3T^{2} \)
89 \( 1 - 26.6iT - 7.92e3T^{2} \)
97 \( 1 + 13.7T + 9.40e3T^{2} \)
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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

−13.22922097362670025780314593733, −12.00776960010613520727335228559, −10.79154181490182152522261972845, −9.653488976942038647755470025708, −8.849420710414752922831875638455, −7.983055220813863670794480895504, −6.51972069822826180201030510663, −5.73340973001260549119152749515, −4.75266765147084019729036843231, −3.33198132617279387522168370872, 0.28983673899925020920387321588, 2.25740468155045085048126689623, 2.97349642446235093781582722881, 4.71839513809229331020567821609, 6.65619874892819358054309787023, 7.07223890743544896870016476012, 8.772795108946971061178702447159, 9.959027363744007731741112602854, 10.52797584347180310303180957067, 11.80619946222670856529526215065

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