Properties

Label 2-201-67.37-c1-0-9
Degree $2$
Conductor $201$
Sign $0.606 + 0.795i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0732 − 0.126i)2-s + 3-s + (0.989 − 1.71i)4-s − 1.65·5-s + (−0.0732 − 0.126i)6-s + (1.22 − 2.11i)7-s − 0.582·8-s + 9-s + (0.120 + 0.209i)10-s + (0.0152 − 0.0263i)11-s + (0.989 − 1.71i)12-s + (−0.0993 − 0.172i)13-s − 0.357·14-s − 1.65·15-s + (−1.93 − 3.35i)16-s + (0.847 + 1.46i)17-s + ⋯
L(s)  = 1  + (−0.0517 − 0.0896i)2-s + 0.577·3-s + (0.494 − 0.856i)4-s − 0.738·5-s + (−0.0298 − 0.0517i)6-s + (0.461 − 0.798i)7-s − 0.205·8-s + 0.333·9-s + (0.0382 + 0.0662i)10-s + (0.00458 − 0.00794i)11-s + (0.285 − 0.494i)12-s + (−0.0275 − 0.0477i)13-s − 0.0955·14-s − 0.426·15-s + (−0.483 − 0.838i)16-s + (0.205 + 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.606 + 0.795i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.606 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26113 - 0.624103i\)
\(L(\frac12)\) \(\approx\) \(1.26113 - 0.624103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
67 \( 1 + (-0.201 - 8.18i)T \)
good2 \( 1 + (0.0732 + 0.126i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 + (-1.22 + 2.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0152 + 0.0263i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0993 + 0.172i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.847 - 1.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.97 - 5.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.79 - 3.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.17 + 3.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.519 - 0.899i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.72 - 2.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.49 - 6.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 + (2.74 - 4.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.67T + 53T^{2} \)
59 \( 1 + 7.66T + 59T^{2} \)
61 \( 1 + (-1.19 - 2.07i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-5.29 + 9.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.00 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.81 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.82 + 6.62i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + (0.0738 + 0.127i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.09474214270731957976740090834, −11.29724083405457983961328382788, −10.33657680368254075400973340019, −9.543648325540699708683046735166, −8.057933853492093896978875473628, −7.44006497087485684435010399879, −6.11368801827226444860149796467, −4.65937422092524327315080307155, −3.34706000030559883175994745061, −1.46335661689208396874154308218, 2.43677869433105711311987098506, 3.58067226623907146239874404239, 5.00128938496402356098746724024, 6.74720178527966375366144201434, 7.66635113894458944230110402239, 8.473124306434552791131247580727, 9.295601297269114382910949812930, 10.91451128533703323431750073183, 11.76079182857890469289654260902, 12.40123037606550857061307511647

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