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Properties

Label 2-201-67.30-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.111 - 0.993i$
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  + (−3.23 + 1.86i)2-s + 1.73i·3-s + (4.95 − 8.58i)4-s − 1.91i·5-s + (−3.23 − 5.59i)6-s + (−11.0 − 6.36i)7-s + 22.0i·8-s − 2.99·9-s + (3.57 + 6.19i)10-s + (14.7 + 8.49i)11-s + (14.8 + 8.58i)12-s + (3.55 − 2.05i)13-s + 47.4·14-s + 3.32·15-s + (−21.3 − 36.9i)16-s + (−4.51 − 7.81i)17-s + ⋯
L(s)  = 1  + (−1.61 + 0.932i)2-s + 0.577i·3-s + (1.23 − 2.14i)4-s − 0.383i·5-s + (−0.538 − 0.932i)6-s + (−1.57 − 0.909i)7-s + 2.75i·8-s − 0.333·9-s + (0.357 + 0.619i)10-s + (1.33 + 0.771i)11-s + (1.23 + 0.715i)12-s + (0.273 − 0.157i)13-s + 3.39·14-s + 0.221·15-s + (−1.33 − 2.30i)16-s + (−0.265 − 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.111 - 0.993i$
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: \((2,\ 201,\ (\ :1),\ -0.111 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.380459 + 0.425633i\)
\(L(\frac12)\) \(\approx\) \(0.380459 + 0.425633i\)
\(L(2)\) 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 \( 1 + (-50.0 - 44.5i)T \)
good2 \( 1 + (3.23 - 1.86i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 1.91iT - 25T^{2} \)
7 \( 1 + (11.0 + 6.36i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-14.7 - 8.49i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.55 + 2.05i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (4.51 + 7.81i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.3 - 23.1i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.5 - 30.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (13.4 - 23.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-10.2 - 5.93i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (3.10 + 5.37i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (4.90 + 2.83i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + 79.5iT - 1.84e3T^{2} \)
47 \( 1 + (2.52 - 4.36i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 77.2iT - 2.80e3T^{2} \)
59 \( 1 - 60.3T + 3.48e3T^{2} \)
61 \( 1 + (-48.2 + 27.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (1.34 - 2.33i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-28.9 - 50.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-37.6 - 21.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (0.640 + 1.10i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 55.9T + 7.92e3T^{2} \)
97 \( 1 + (33.3 - 19.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.29390601592274976452866503557, −10.96480686324120719774682601763, −9.969033137043134098371662319590, −9.503125994890596155966491298277, −8.783164265818525001514123216981, −7.27738366221359350586622572517, −6.75565247202201854127478121710, −5.51784491953921277628918316404, −3.68207321240221799620303423940, −1.08036944674572430719216499590, 0.70486369230845158481375452709, 2.53025642142353087017906213707, 3.38782683726320034625387753873, 6.42484550786823751343174791599, 6.80669470939926489999432507090, 8.422609244730885018971739090868, 9.078455936915813396295436304635, 9.758926573819418062464782899755, 11.05126093525846989508744537053, 11.66434352545417502956398567322

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