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Properties

Label 2-201-201.200-c1-0-10
Degree $2$
Conductor $201$
Sign $0.774 - 0.632i$
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  + 2.09·2-s + (−1.16 + 1.28i)3-s + 2.38·4-s + 1.47·5-s + (−2.44 + 2.68i)6-s + 2.11i·7-s + 0.798·8-s + (−0.281 − 2.98i)9-s + 3.09·10-s + 1.98·11-s + (−2.77 + 3.05i)12-s − 0.757i·13-s + 4.43i·14-s + (−1.72 + 1.89i)15-s − 3.09·16-s − 4.64i·17-s + ⋯
L(s)  = 1  + 1.48·2-s + (−0.673 + 0.739i)3-s + 1.19·4-s + 0.660·5-s + (−0.996 + 1.09i)6-s + 0.800i·7-s + 0.282·8-s + (−0.0936 − 0.995i)9-s + 0.977·10-s + 0.598·11-s + (−0.801 + 0.880i)12-s − 0.210i·13-s + 1.18i·14-s + (−0.444 + 0.488i)15-s − 0.772·16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01567 + 0.718661i\)
\(L(\frac12)\) \(\approx\) \(2.01567 + 0.718661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 - 1.28i)T \)
67 \( 1 + (-8.09 - 1.20i)T \)
good2 \( 1 - 2.09T + 2T^{2} \)
5 \( 1 - 1.47T + 5T^{2} \)
7 \( 1 - 2.11iT - 7T^{2} \)
11 \( 1 - 1.98T + 11T^{2} \)
13 \( 1 + 0.757iT - 13T^{2} \)
17 \( 1 + 4.64iT - 17T^{2} \)
19 \( 1 + 1.49T + 19T^{2} \)
23 \( 1 + 3.56iT - 23T^{2} \)
29 \( 1 + 2.91iT - 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 3.85T + 37T^{2} \)
41 \( 1 + 3.03T + 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 - 5.79iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 14.2iT - 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
71 \( 1 + 4.38iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 17.6iT - 79T^{2} \)
83 \( 1 + 5.31iT - 83T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + 7.55iT - 97T^{2} \)
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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

−12.39380805949765150193302313688, −11.92049648675857254088065020220, −10.93931917492429819540774232602, −9.713926736591871076547243532186, −8.864465736249618275842881590495, −6.75655506146804001859374420058, −5.88026187296419329668294204013, −5.15767458302728102532633062293, −4.08587404700328560041364707338, −2.67410437813553966203515815590, 1.86415770901429275964449738495, 3.74061805522464458864914007246, 4.92902559242463142325646638690, 6.08828543004255678773501930040, 6.60673778769647911376793289088, 7.914090106958505160733689215372, 9.604256127849149452818850482725, 10.89051699693194430810265331011, 11.61595874621975835218799390799, 12.69066010601471145115428252805

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