Properties

Label 2-201-201.38-c1-0-19
Degree $2$
Conductor $201$
Sign $-0.592 + 0.805i$
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  + (1.30 − 2.25i)2-s + (1.69 + 0.341i)3-s + (−2.40 − 4.15i)4-s − 3.98·5-s + (2.98 − 3.38i)6-s + (2.31 − 1.33i)7-s − 7.30·8-s + (2.76 + 1.16i)9-s + (−5.19 + 9.00i)10-s + (2.06 + 3.57i)11-s + (−2.65 − 7.88i)12-s + (1.09 + 0.634i)13-s − 6.97i·14-s + (−6.76 − 1.36i)15-s + (−4.72 + 8.18i)16-s + (1.45 + 0.841i)17-s + ⋯
L(s)  = 1  + (0.922 − 1.59i)2-s + (0.980 + 0.197i)3-s + (−1.20 − 2.07i)4-s − 1.78·5-s + (1.21 − 1.38i)6-s + (0.875 − 0.505i)7-s − 2.58·8-s + (0.922 + 0.387i)9-s + (−1.64 + 2.84i)10-s + (0.621 + 1.07i)11-s + (−0.766 − 2.27i)12-s + (0.305 + 0.176i)13-s − 1.86i·14-s + (−1.74 − 0.352i)15-s + (−1.18 + 2.04i)16-s + (0.353 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886424 - 1.75267i\)
\(L(\frac12)\) \(\approx\) \(0.886424 - 1.75267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.341i)T \)
67 \( 1 + (3.22 + 7.52i)T \)
good2 \( 1 + (-1.30 + 2.25i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.98T + 5T^{2} \)
7 \( 1 + (-2.31 + 1.33i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 - 0.634i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 0.841i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.716 + 1.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.35 + 2.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.28 - 0.743i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.70 - 4.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0952 - 0.165i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.545 + 0.944i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.84iT - 43T^{2} \)
47 \( 1 + (5.34 - 3.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.507T + 53T^{2} \)
59 \( 1 - 5.37iT - 59T^{2} \)
61 \( 1 + (-1.26 - 0.727i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-13.2 + 7.64i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.84 - 3.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.89 + 1.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.39 + 3.69i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + (8.00 + 4.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.16027126276084070305490155760, −11.26318213591385050064621783107, −10.55355845547927710909354986484, −9.387595918091439203130721337551, −8.245113446923282283311055792782, −7.21934712948757187089118432926, −4.72542370624534811334526387483, −4.16371623186934560526693659415, −3.35165149454580709462618117116, −1.64392635809863080551628809612, 3.49395640891013362390890818069, 4.03760305216571512980836999158, 5.43161823471687333347278173056, 6.84008327482028724240381819500, 7.970115061130450068162596675941, 8.076499976090055369694471773972, 9.053141711396257858534808692517, 11.37901363029497251082162255895, 12.09093080183365723494952512632, 13.09245195571964096431417734966

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