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Properties

Label 2-201-67.29-c1-0-8
Degree $2$
Conductor $201$
Sign $-0.311 + 0.950i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + 3-s + (−0.999 − 1.73i)4-s − 4·5-s + (−1 + 1.73i)6-s + (−2 − 3.46i)7-s + 9-s + (4 − 6.92i)10-s + (−1 − 1.73i)11-s + (−0.999 − 1.73i)12-s + (−2.5 + 4.33i)13-s + 7.99·14-s − 4·15-s + (1.99 − 3.46i)16-s + (−3 + 5.19i)17-s + (−1 + 1.73i)18-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + 0.577·3-s + (−0.499 − 0.866i)4-s − 1.78·5-s + (−0.408 + 0.707i)6-s + (−0.755 − 1.30i)7-s + 0.333·9-s + (1.26 − 2.19i)10-s + (−0.301 − 0.522i)11-s + (−0.288 − 0.499i)12-s + (−0.693 + 1.20i)13-s + 2.13·14-s − 1.03·15-s + (0.499 − 0.866i)16-s + (−0.727 + 1.26i)17-s + (−0.235 + 0.408i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.311 + 0.950i$
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: \(1\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 7.79i)T \)
good2 \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 4T + 5T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 11T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.5 - 4.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.23086165806292277445098787425, −10.99683482385521622623887595904, −9.954886656306415136688244889497, −8.618949779666699666392606425381, −8.139342244651102493774331279344, −7.08755571780296197078659705224, −6.64597990405204972066798528472, −4.41190833279596213261213130056, −3.50320329171872096876636162256, 0, 2.66000831798591524831404248579, 3.29156844233987671523602547622, 4.95639132542589955525772548039, 7.02883681637147429131239202300, 8.094793280557917185855673081204, 8.961778337377802661039492651309, 9.693443785373769386657026463167, 10.93193604425546946009139243392, 11.74873481621477047475117588346

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