Properties

Label 2-201-201.161-c3-0-46
Degree $2$
Conductor $201$
Sign $0.770 + 0.637i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.72 + 2.15i)3-s + (1.13 − 7.91i)4-s + (11.6 − 10.0i)7-s + (17.6 + 20.4i)9-s + (22.4 − 34.9i)12-s + (6.52 − 10.1i)13-s + (−61.4 − 18.0i)16-s + (108. − 124. i)19-s + (76.5 − 22.4i)21-s + (−105. − 67.5i)25-s + (39.5 + 134. i)27-s + (−66.4 − 103. i)28-s + (127. + 198. i)31-s + (181. − 116. i)36-s + 143.·37-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)3-s + (0.142 − 0.989i)4-s + (0.626 − 0.543i)7-s + (0.654 + 0.755i)9-s + (0.540 − 0.841i)12-s + (0.139 − 0.216i)13-s + (−0.959 − 0.281i)16-s + (1.30 − 1.50i)19-s + (0.795 − 0.233i)21-s + (−0.841 − 0.540i)25-s + (0.281 + 0.959i)27-s + (−0.448 − 0.697i)28-s + (0.740 + 1.15i)31-s + (0.841 − 0.540i)36-s + 0.637·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.770 + 0.637i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.46428 - 0.887514i\)
\(L(\frac12)\) \(\approx\) \(2.46428 - 0.887514i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.72 - 2.15i)T \)
67 \( 1 + (-193. - 513. i)T \)
good2 \( 1 + (-1.13 + 7.91i)T^{2} \)
5 \( 1 + (105. + 67.5i)T^{2} \)
7 \( 1 + (-11.6 + 10.0i)T + (48.8 - 339. i)T^{2} \)
11 \( 1 + (1.11e3 + 719. i)T^{2} \)
13 \( 1 + (-6.52 + 10.1i)T + (-912. - 1.99e3i)T^{2} \)
17 \( 1 + (4.71e3 - 1.38e3i)T^{2} \)
19 \( 1 + (-108. + 124. i)T + (-976. - 6.78e3i)T^{2} \)
23 \( 1 + (7.96e3 + 9.19e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-127. - 198. i)T + (-1.23e4 + 2.70e4i)T^{2} \)
37 \( 1 - 143.T + 5.06e4T^{2} \)
41 \( 1 + (-6.61e4 + 1.94e4i)T^{2} \)
43 \( 1 + (-96.5 + 13.8i)T + (7.62e4 - 2.23e4i)T^{2} \)
47 \( 1 + (6.79e4 + 7.84e4i)T^{2} \)
53 \( 1 + (-1.42e5 - 4.19e4i)T^{2} \)
59 \( 1 + (-8.53e4 + 1.86e5i)T^{2} \)
61 \( 1 + (156. + 531. i)T + (-1.90e5 + 1.22e5i)T^{2} \)
71 \( 1 + (3.43e5 + 1.00e5i)T^{2} \)
73 \( 1 + (1.19e3 - 351. i)T + (3.27e5 - 2.10e5i)T^{2} \)
79 \( 1 + (754. - 1.17e3i)T + (-2.04e5 - 4.48e5i)T^{2} \)
83 \( 1 + (-4.81e5 - 3.09e5i)T^{2} \)
89 \( 1 + (4.61e5 - 5.32e5i)T^{2} \)
97 \( 1 - 1.77e3iT - 9.12e5T^{2} \)
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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

−11.61319739384845226843321472284, −10.74220560566686446754589329728, −9.906947794607944767195770401437, −9.062050812822950387041777913071, −7.893362268911845402201503254980, −6.86819770905064173115550806280, −5.30077545943877556901095558762, −4.35598088846051187259204263038, −2.72159204847951802476787886950, −1.17950517494264521580499045623, 1.77087436685935864981232633078, 3.06810282134756963647375859535, 4.19357240487032058596636134192, 5.95399873626036341225709415916, 7.44027094431144237997940026543, 7.963986625241423713847589223312, 8.891130740696405260538033963917, 9.886022045974297538408445530002, 11.57599029450035145695544509356, 12.04535490328697813685676911863

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