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Properties

Label 2-201-201.38-c1-0-15
Degree $2$
Conductor $201$
Sign $0.725 + 0.688i$
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  + (0.451 − 0.782i)2-s + (1.52 − 0.820i)3-s + (0.591 + 1.02i)4-s − 1.39·5-s + (0.0470 − 1.56i)6-s + (−0.0330 + 0.0190i)7-s + 2.87·8-s + (1.65 − 2.50i)9-s + (−0.631 + 1.09i)10-s + (−1.29 − 2.24i)11-s + (1.74 + 1.07i)12-s + (2.41 + 1.39i)13-s + 0.0345i·14-s + (−2.13 + 1.14i)15-s + (0.115 − 0.199i)16-s + (−3.52 − 2.03i)17-s + ⋯
L(s)  = 1  + (0.319 − 0.553i)2-s + (0.880 − 0.473i)3-s + (0.295 + 0.512i)4-s − 0.625·5-s + (0.0192 − 0.638i)6-s + (−0.0125 + 0.00721i)7-s + 1.01·8-s + (0.551 − 0.834i)9-s + (−0.199 + 0.345i)10-s + (−0.390 − 0.676i)11-s + (0.503 + 0.311i)12-s + (0.670 + 0.387i)13-s + 0.00922i·14-s + (−0.550 + 0.296i)15-s + (0.0288 − 0.0498i)16-s + (−0.854 − 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.725 + 0.688i$
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: \((2,\ 201,\ (\ :1/2),\ 0.725 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66056 - 0.662828i\)
\(L(\frac12)\) \(\approx\) \(1.66056 - 0.662828i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.52 + 0.820i)T \)
67 \( 1 + (7.49 + 3.28i)T \)
good2 \( 1 + (-0.451 + 0.782i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.39T + 5T^{2} \)
7 \( 1 + (0.0330 - 0.0190i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.29 + 2.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.41 - 1.39i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.52 + 2.03i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.93 - 6.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.808 - 0.466i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.19 - 4.15i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.33 + 1.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.92 + 6.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.99 - 5.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.14iT - 43T^{2} \)
47 \( 1 + (2.55 - 1.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.55T + 53T^{2} \)
59 \( 1 + 0.806iT - 59T^{2} \)
61 \( 1 + (-0.740 - 0.427i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-3.83 + 2.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.27 + 3.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.0 + 6.94i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.8 - 6.28i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + (-1.73 - 1.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.45223578321729963217630358577, −11.48468512860308332755086862491, −10.70406724512631947330801520229, −9.185849754857654103032575325867, −8.155977060192509256012081370227, −7.53748176029953522469843356500, −6.26230911971081906382990408393, −4.17152311286245049581552750397, −3.38426360553811283502504202356, −1.99122546934302984027425971324, 2.27672946008844617423201767639, 4.03684210993074003694653689534, 4.94427880047620592556036198523, 6.45804092037287990360776614171, 7.51721731154708028307641267768, 8.412162808742963910040131622786, 9.583687041766407898490386634035, 10.63677662722250569559759306538, 11.33060949172884889487979363903, 13.08004126974729189715153208304

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