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Properties

Label 2-201-67.30-c2-0-0
Degree $2$
Conductor $201$
Sign $-0.808 - 0.588i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.658 − 0.380i)2-s − 1.73i·3-s + (−1.71 + 2.96i)4-s + 4.20i·5-s + (−0.658 − 1.14i)6-s + (−7.58 − 4.37i)7-s + 5.64i·8-s − 2.99·9-s + (1.59 + 2.76i)10-s + (−10.7 − 6.21i)11-s + (5.13 + 2.96i)12-s + (−15.3 + 8.88i)13-s − 6.65·14-s + 7.28·15-s + (−4.70 − 8.14i)16-s + (1.47 + 2.55i)17-s + ⋯
L(s)  = 1  + (0.329 − 0.190i)2-s − 0.577i·3-s + (−0.427 + 0.740i)4-s + 0.841i·5-s + (−0.109 − 0.190i)6-s + (−1.08 − 0.625i)7-s + 0.705i·8-s − 0.333·9-s + (0.159 + 0.276i)10-s + (−0.978 − 0.565i)11-s + (0.427 + 0.246i)12-s + (−1.18 + 0.683i)13-s − 0.475·14-s + 0.485·15-s + (−0.293 − 0.508i)16-s + (0.0868 + 0.150i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.808 - 0.588i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.808 - 0.588i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.132077 + 0.406204i\)
\(L(\frac12)\) \(\approx\) \(0.132077 + 0.406204i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (66.6 - 6.80i)T \)
good2 \( 1 + (-0.658 + 0.380i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 4.20iT - 25T^{2} \)
7 \( 1 + (7.58 + 4.37i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.7 + 6.21i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (15.3 - 8.88i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.47 - 2.55i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.76 - 6.51i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.18 - 5.51i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (14.0 - 24.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-36.3 - 20.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-18.0 - 31.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (61.7 + 35.6i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + 24.6iT - 1.84e3T^{2} \)
47 \( 1 + (-12.1 + 21.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 45.4iT - 2.80e3T^{2} \)
59 \( 1 + 15.3T + 3.48e3T^{2} \)
61 \( 1 + (64.1 - 37.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-23.3 + 40.4i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-65.0 - 112. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-96.8 - 55.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-29.2 - 50.7i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 81.1T + 7.92e3T^{2} \)
97 \( 1 + (-17.9 + 10.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.64363393048859746136523894305, −11.90904972690459177960687207191, −10.72109276398066717729540815387, −9.788004385964762855945541251663, −8.465746935806542588226706678935, −7.35373551055469343176798617184, −6.69479616728360407117888031617, −5.12599972803132006677795245671, −3.52087505996838271427910050241, −2.68308295779494701947983128621, 0.20668030388649579423080382476, 2.78392739965416788215279638968, 4.58129895961214749125991889742, 5.23134233078214581858896227841, 6.26499996218432968904437501160, 7.83712606251095639367915383370, 9.247239040120119122846166318992, 9.692373823039428898386663553673, 10.54020355648899056000420164058, 12.15240626655953039228558273593

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