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Properties

Label 2-201-67.29-c1-0-2
Degree $2$
Conductor $201$
Sign $0.470 - 0.882i$
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  + (−0.330 + 0.571i)2-s − 3-s + (0.782 + 1.35i)4-s + 3.22·5-s + (0.330 − 0.571i)6-s + (−1.02 − 1.76i)7-s − 2.35·8-s + 9-s + (−1.06 + 1.84i)10-s + (2.63 + 4.56i)11-s + (−0.782 − 1.35i)12-s + (0.956 − 1.65i)13-s + 1.34·14-s − 3.22·15-s + (−0.787 + 1.36i)16-s + (−1.83 + 3.18i)17-s + ⋯
L(s)  = 1  + (−0.233 + 0.404i)2-s − 0.577·3-s + (0.391 + 0.677i)4-s + 1.44·5-s + (0.134 − 0.233i)6-s + (−0.386 − 0.668i)7-s − 0.832·8-s + 0.333·9-s + (−0.336 + 0.583i)10-s + (0.794 + 1.37i)11-s + (−0.225 − 0.391i)12-s + (0.265 − 0.459i)13-s + 0.360·14-s − 0.832·15-s + (−0.196 + 0.340i)16-s + (−0.445 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.470 - 0.882i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.996358 + 0.598059i\)
\(L(\frac12)\) \(\approx\) \(0.996358 + 0.598059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 8.10i)T \)
good2 \( 1 + (0.330 - 0.571i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 3.22T + 5T^{2} \)
7 \( 1 + (1.02 + 1.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.63 - 4.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.956 + 1.65i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.83 - 3.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.564 + 0.977i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.160 + 0.277i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.24 - 5.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.63 + 6.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.40 + 7.63i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.86 + 6.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + (4.74 + 8.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 + (-2.99 + 5.18i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (1.02 + 1.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.23 - 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.61 - 2.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0476 - 0.0825i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.17T + 89T^{2} \)
97 \( 1 + (-5.80 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.79685030874298890177042158062, −11.70536843166010275431125525350, −10.47573501339236087365774725669, −9.751625323794874194127409544103, −8.702048389518274840173210083533, −7.13083328414549384900671266401, −6.63364644891778183012264408252, −5.52927907720217853916829779049, −3.90619143216468140521000985989, −2.01070924205769158696433545727, 1.41238969206017423308839418748, 2.88684668420320182982244949817, 5.16820664137585223136035283266, 6.21865068415002384085111817446, 6.45603644514140780397705602103, 8.772949190199536515350572027288, 9.510954512903707502613300418104, 10.26787950755966782510170999232, 11.36505966047715408582939273659, 11.90673214107538562927727855386

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