Properties

Label 2-201-67.37-c1-0-5
Degree $2$
Conductor $201$
Sign $0.963 - 0.268i$
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  + (0.527 + 0.913i)2-s − 3-s + (0.443 − 0.768i)4-s + 0.832·5-s + (−0.527 − 0.913i)6-s + (1.13 − 1.95i)7-s + 3.04·8-s + 9-s + (0.439 + 0.760i)10-s + (−0.713 + 1.23i)11-s + (−0.443 + 0.768i)12-s + (0.308 + 0.535i)13-s + 2.38·14-s − 0.832·15-s + (0.718 + 1.24i)16-s + (2.57 + 4.46i)17-s + ⋯
L(s)  = 1  + (0.372 + 0.645i)2-s − 0.577·3-s + (0.221 − 0.384i)4-s + 0.372·5-s + (−0.215 − 0.372i)6-s + (0.427 − 0.739i)7-s + 1.07·8-s + 0.333·9-s + (0.138 + 0.240i)10-s + (−0.215 + 0.372i)11-s + (−0.128 + 0.221i)12-s + (0.0856 + 0.148i)13-s + 0.637·14-s − 0.215·15-s + (0.179 + 0.311i)16-s + (0.625 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.963 - 0.268i$
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: \((2,\ 201,\ (\ :1/2),\ 0.963 - 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43778 + 0.196908i\)
\(L(\frac12)\) \(\approx\) \(1.43778 + 0.196908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.67 + 2.84i)T \)
good2 \( 1 + (-0.527 - 0.913i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 0.832T + 5T^{2} \)
7 \( 1 + (-1.13 + 1.95i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.713 - 1.23i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.308 - 0.535i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.57 - 4.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.939 + 1.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.55 + 2.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.962 - 1.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.286 - 0.495i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.67 + 4.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.25 - 7.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 + (-0.624 + 1.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.80T + 53T^{2} \)
59 \( 1 - 7.75T + 59T^{2} \)
61 \( 1 + (1.96 + 3.41i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (4.64 - 8.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.88 - 13.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.745 - 1.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.355 + 0.616i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.514T + 89T^{2} \)
97 \( 1 + (-0.590 - 1.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.66135857806901598712729360475, −11.38890282678737049199447018499, −10.51957034365673642179401594434, −9.883816428013626607380392135079, −8.157574411713261248393233003599, −7.12586445297058036069078827403, −6.20988613533507674950079249491, −5.23575718416883969872403573802, −4.16055091844605923644520770839, −1.65598373386351520776530467591, 1.94963898123484189169981767258, 3.39885774991435842115899464408, 4.92926382415114839823674496285, 5.88950829992771066964229129334, 7.31794066406767509446388437883, 8.352450813406031258266079578036, 9.748729654866114880701003647776, 10.69819131390892445005506616784, 11.82360762970424889448237670542, 11.97378857427497894349028721175

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