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Properties

Label 2-201-201.38-c1-0-6
Degree $2$
Conductor $201$
Sign $0.984 - 0.175i$
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  + (0.577 − 0.999i)2-s + (−1.56 + 0.750i)3-s + (0.333 + 0.578i)4-s + 2.29·5-s + (−0.150 + 1.99i)6-s + (−3.24 + 1.87i)7-s + 3.07·8-s + (1.87 − 2.34i)9-s + (1.32 − 2.29i)10-s + (1.40 + 2.43i)11-s + (−0.954 − 0.651i)12-s + (5.39 + 3.11i)13-s + 4.32i·14-s + (−3.57 + 1.72i)15-s + (1.10 − 1.92i)16-s + (−4.98 − 2.87i)17-s + ⋯
L(s)  = 1  + (0.408 − 0.706i)2-s + (−0.901 + 0.433i)3-s + (0.166 + 0.289i)4-s + 1.02·5-s + (−0.0615 + 0.813i)6-s + (−1.22 + 0.708i)7-s + 1.08·8-s + (0.624 − 0.781i)9-s + (0.418 − 0.724i)10-s + (0.423 + 0.733i)11-s + (−0.275 − 0.188i)12-s + (1.49 + 0.864i)13-s + 1.15i·14-s + (−0.923 + 0.444i)15-s + (0.277 − 0.480i)16-s + (−1.20 − 0.697i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.984 - 0.175i$
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: \((2,\ 201,\ (\ :1/2),\ 0.984 - 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34023 + 0.118546i\)
\(L(\frac12)\) \(\approx\) \(1.34023 + 0.118546i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 - 0.750i)T \)
67 \( 1 + (-6.89 + 4.40i)T \)
good2 \( 1 + (-0.577 + 0.999i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.29T + 5T^{2} \)
7 \( 1 + (3.24 - 1.87i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.39 - 3.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.98 + 2.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.82 + 4.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.01 + 2.31i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.50 + 0.867i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.38 - 1.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.522 - 0.904i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.05 + 3.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 7.81iT - 43T^{2} \)
47 \( 1 + (8.61 - 4.97i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.49T + 53T^{2} \)
59 \( 1 + 5.92iT - 59T^{2} \)
61 \( 1 + (9.10 + 5.25i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-7.40 + 4.27i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.102 + 0.177i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.9 + 6.31i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0300 + 0.0173i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.401iT - 89T^{2} \)
97 \( 1 + (6.34 + 3.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.34384079110951233928955140475, −11.59014728534346290633075331420, −10.73630922329834999488256681109, −9.632996192368480296410733398119, −9.054193978431622464760740904878, −6.78558913803517354279904570503, −6.30103427032206595152524272185, −4.87293793497201413292725053805, −3.62955631975087754690969126300, −2.08614964702318129429429310914, 1.40904200047766924386816328880, 3.84816968351118893236956406433, 5.64819189205424230441772964537, 6.11996532288955687849529724902, 6.70962794797201236871464340410, 8.096005640711100768822914403419, 9.802629020385762869949992933601, 10.47468951341626958726327797627, 11.30090136683590382354446125506, 12.87029151636793368133788736579

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