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Properties

Label 2-201-67.37-c1-0-1
Degree $2$
Conductor $201$
Sign $-0.998 - 0.0596i$
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  + (1.03 + 1.79i)2-s − 3-s + (−1.13 + 1.96i)4-s − 3.33·5-s + (−1.03 − 1.79i)6-s + (−2.29 + 3.97i)7-s − 0.563·8-s + 9-s + (−3.45 − 5.97i)10-s + (0.625 − 1.08i)11-s + (1.13 − 1.96i)12-s + (−0.156 − 0.271i)13-s − 9.48·14-s + 3.33·15-s + (1.69 + 2.92i)16-s + (1.78 + 3.09i)17-s + ⋯
L(s)  = 1  + (0.730 + 1.26i)2-s − 0.577·3-s + (−0.568 + 0.984i)4-s − 1.49·5-s + (−0.421 − 0.730i)6-s + (−0.867 + 1.50i)7-s − 0.199·8-s + 0.333·9-s + (−1.09 − 1.89i)10-s + (0.188 − 0.326i)11-s + (0.328 − 0.568i)12-s + (−0.0433 − 0.0751i)13-s − 2.53·14-s + 0.862·15-s + (0.422 + 0.731i)16-s + (0.432 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.998 - 0.0596i$
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: \((2,\ 201,\ (\ :1/2),\ -0.998 - 0.0596i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0278647 + 0.933222i\)
\(L(\frac12)\) \(\approx\) \(0.0278647 + 0.933222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.32 - 5.18i)T \)
good2 \( 1 + (-1.03 - 1.79i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.33T + 5T^{2} \)
7 \( 1 + (2.29 - 3.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.625 + 1.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.156 + 0.271i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.78 - 3.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.95 - 5.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.56 + 4.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.73 + 8.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.62 - 2.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.61 - 4.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.63 - 2.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 6.40T + 43T^{2} \)
47 \( 1 + (5.68 - 9.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.86T + 53T^{2} \)
59 \( 1 - 3.85T + 59T^{2} \)
61 \( 1 + (2.79 + 4.84i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-5.36 + 9.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.36 + 5.83i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.22 - 3.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.62 - 9.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.04T + 89T^{2} \)
97 \( 1 + (3.31 + 5.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.62800348593829254086405006878, −12.33039961054806626152121505436, −11.37100958840655144101789485891, −9.934881133815873173486269060031, −8.391653129538931913364385910960, −7.80878599327025970085083386965, −6.38927164796203452543293097219, −5.89642225479223442103815069263, −4.58840930808542878934965272017, −3.41415561635671566346871491767, 0.72231161030691043697910990430, 3.28446480564681118145906661137, 4.01053887700997441689497100411, 4.98673106834942589343039714149, 7.01558973432644595129820046950, 7.53309652050567391995986628458, 9.512492114827583059940156053128, 10.45615772138458586172288779890, 11.26263107822476695548320826124, 11.86490487449600990712265767310

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