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Properties

Label 2-201-3.2-c2-0-21
Degree $2$
Conductor $201$
Sign $0.898 + 0.438i$
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  + 3.74i·2-s + (−2.69 − 1.31i)3-s − 10.0·4-s + 4.74i·5-s + (4.93 − 10.1i)6-s − 4.78·7-s − 22.6i·8-s + (5.53 + 7.09i)9-s − 17.7·10-s − 18.6i·11-s + (27.0 + 13.2i)12-s − 4.14·13-s − 17.9i·14-s + (6.25 − 12.7i)15-s + 44.8·16-s + 26.7i·17-s + ⋯
L(s)  = 1  + 1.87i·2-s + (−0.898 − 0.438i)3-s − 2.51·4-s + 0.949i·5-s + (0.822 − 1.68i)6-s − 0.682·7-s − 2.83i·8-s + (0.614 + 0.788i)9-s − 1.77·10-s − 1.69i·11-s + (2.25 + 1.10i)12-s − 0.318·13-s − 1.28i·14-s + (0.416 − 0.853i)15-s + 2.80·16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.898 + 0.438i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.153589 - 0.0355079i\)
\(L(\frac12)\) \(\approx\) \(0.153589 - 0.0355079i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.69 + 1.31i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 3.74iT - 4T^{2} \)
5 \( 1 - 4.74iT - 25T^{2} \)
7 \( 1 + 4.78T + 49T^{2} \)
11 \( 1 + 18.6iT - 121T^{2} \)
13 \( 1 + 4.14T + 169T^{2} \)
17 \( 1 - 26.7iT - 289T^{2} \)
19 \( 1 - 9.76T + 361T^{2} \)
23 \( 1 + 28.7iT - 529T^{2} \)
29 \( 1 + 22.6iT - 841T^{2} \)
31 \( 1 + 41.0T + 961T^{2} \)
37 \( 1 + 24.6T + 1.36e3T^{2} \)
41 \( 1 + 51.5iT - 1.68e3T^{2} \)
43 \( 1 + 68.1T + 1.84e3T^{2} \)
47 \( 1 - 29.5iT - 2.20e3T^{2} \)
53 \( 1 + 52.1iT - 2.80e3T^{2} \)
59 \( 1 + 32.9iT - 3.48e3T^{2} \)
61 \( 1 + 72.5T + 3.72e3T^{2} \)
71 \( 1 - 48.2iT - 5.04e3T^{2} \)
73 \( 1 + 6.54T + 5.32e3T^{2} \)
79 \( 1 - 93.9T + 6.24e3T^{2} \)
83 \( 1 - 38.8iT - 6.88e3T^{2} \)
89 \( 1 - 33.5iT - 7.92e3T^{2} \)
97 \( 1 - 87.5T + 9.40e3T^{2} \)
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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

−12.53237334650875038992809461511, −11.00986856925801433562175707297, −10.14276923121051689422469190843, −8.708516774744703605352371737041, −7.77040946901993679127883775444, −6.62845969841823561520689526004, −6.26301617781815307685587387121, −5.31593967259637565065129381544, −3.65525084759371594007776469158, −0.10699474383194849592815752752, 1.48749191238539293891837540713, 3.33488383278611358776728978799, 4.71478380837335912599475544380, 5.15609911829192014778684569276, 7.23436082864440485982323123422, 9.266392382208998102030462163750, 9.494204200741434132963720818795, 10.34500935151900175669169750968, 11.53679515679799788101833174860, 12.14842196258613541859890462722

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