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Properties

Label 2-201-3.2-c2-0-28
Degree $2$
Conductor $201$
Sign $-0.151 + 0.988i$
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  − 1.81i·2-s + (0.454 − 2.96i)3-s + 0.717·4-s + 6.02i·5-s + (−5.37 − 0.824i)6-s + 13.4·7-s − 8.54i·8-s + (−8.58 − 2.69i)9-s + 10.9·10-s − 4.38i·11-s + (0.326 − 2.12i)12-s + 11.7·13-s − 24.4i·14-s + (17.8 + 2.74i)15-s − 12.6·16-s + 1.06i·17-s + ⋯
L(s)  = 1  − 0.905i·2-s + (0.151 − 0.988i)3-s + 0.179·4-s + 1.20i·5-s + (−0.895 − 0.137i)6-s + 1.92·7-s − 1.06i·8-s + (−0.953 − 0.299i)9-s + 1.09·10-s − 0.398i·11-s + (0.0271 − 0.177i)12-s + 0.901·13-s − 1.74i·14-s + (1.19 + 0.182i)15-s − 0.788·16-s + 0.0629i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.151 + 0.988i$
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: \((2,\ 201,\ (\ :1),\ -0.151 + 0.988i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34908 - 1.57187i\)
\(L(\frac12)\) \(\approx\) \(1.34908 - 1.57187i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.454 + 2.96i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 + 1.81iT - 4T^{2} \)
5 \( 1 - 6.02iT - 25T^{2} \)
7 \( 1 - 13.4T + 49T^{2} \)
11 \( 1 + 4.38iT - 121T^{2} \)
13 \( 1 - 11.7T + 169T^{2} \)
17 \( 1 - 1.06iT - 289T^{2} \)
19 \( 1 + 26.2T + 361T^{2} \)
23 \( 1 - 15.8iT - 529T^{2} \)
29 \( 1 - 3.83iT - 841T^{2} \)
31 \( 1 + 59.1T + 961T^{2} \)
37 \( 1 - 40.7T + 1.36e3T^{2} \)
41 \( 1 + 20.7iT - 1.68e3T^{2} \)
43 \( 1 - 2.22T + 1.84e3T^{2} \)
47 \( 1 + 6.59iT - 2.20e3T^{2} \)
53 \( 1 + 36.2iT - 2.80e3T^{2} \)
59 \( 1 - 69.4iT - 3.48e3T^{2} \)
61 \( 1 + 75.2T + 3.72e3T^{2} \)
71 \( 1 - 133. iT - 5.04e3T^{2} \)
73 \( 1 - 31.0T + 5.32e3T^{2} \)
79 \( 1 - 21.0T + 6.24e3T^{2} \)
83 \( 1 + 30.9iT - 6.88e3T^{2} \)
89 \( 1 - 148. iT - 7.92e3T^{2} \)
97 \( 1 + 50.4T + 9.40e3T^{2} \)
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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

−11.61766732601524394136939061077, −11.07680960967950101975592357704, −10.68422191492157725576177680580, −8.845242509681454573018502031946, −7.81212995523131375002076899346, −6.93983985176952878124986916772, −5.81755546961010863434736182432, −3.81324043528913700291218876026, −2.42785030366765990015787949117, −1.47138155132443474721964915287, 1.90542050884418482974254391550, 4.36449686548982952696897351912, 4.97042392607597039218928597096, 5.98875583862359694176901966836, 7.75693236150367833189524352292, 8.443080683711685207814789112453, 9.044102240396599918158539348092, 10.80991358194549700082234083535, 11.20763025519986330104374903722, 12.42858756451933286503000019990

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