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Properties

Label 2-201-201.119-c3-0-19
Degree $2$
Conductor $201$
Sign $0.988 + 0.154i$
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  + (−1.46 − 4.98i)3-s + (−3.32 + 7.27i)4-s + (−2.83 − 4.40i)7-s + (−22.7 + 14.5i)9-s + (41.1 + 5.91i)12-s + (89.3 + 12.8i)13-s + (−41.9 − 48.3i)16-s + (82.1 + 52.7i)19-s + (−17.8 + 20.5i)21-s + (17.7 − 123. i)25-s + (106. + 91.8i)27-s + (41.5 − 5.96i)28-s + (331. − 47.6i)31-s + (−30.7 − 213. i)36-s + 447.·37-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)3-s + (−0.415 + 0.909i)4-s + (−0.153 − 0.238i)7-s + (−0.841 + 0.540i)9-s + (0.989 + 0.142i)12-s + (1.90 + 0.274i)13-s + (−0.654 − 0.755i)16-s + (0.991 + 0.637i)19-s + (−0.185 + 0.213i)21-s + (0.142 − 0.989i)25-s + (0.755 + 0.654i)27-s + (0.280 − 0.0402i)28-s + (1.92 − 0.276i)31-s + (−0.142 − 0.989i)36-s + 1.98·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.988 + 0.154i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.988 + 0.154i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43340 - 0.111197i\)
\(L(\frac12)\) \(\approx\) \(1.43340 - 0.111197i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 + 4.98i)T \)
67 \( 1 + (386. - 388. i)T \)
good2 \( 1 + (3.32 - 7.27i)T^{2} \)
5 \( 1 + (-17.7 + 123. i)T^{2} \)
7 \( 1 + (2.83 + 4.40i)T + (-142. + 312. i)T^{2} \)
11 \( 1 + (-189. + 1.31e3i)T^{2} \)
13 \( 1 + (-89.3 - 12.8i)T + (2.10e3 + 618. i)T^{2} \)
17 \( 1 + (3.21e3 - 3.71e3i)T^{2} \)
19 \( 1 + (-82.1 - 52.7i)T + (2.84e3 + 6.23e3i)T^{2} \)
23 \( 1 + (-1.02e4 + 6.57e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-331. + 47.6i)T + (2.85e4 - 8.39e3i)T^{2} \)
37 \( 1 - 447.T + 5.06e4T^{2} \)
41 \( 1 + (-4.51e4 + 5.20e4i)T^{2} \)
43 \( 1 + (496. - 226. i)T + (5.20e4 - 6.00e4i)T^{2} \)
47 \( 1 + (-8.73e4 + 5.61e4i)T^{2} \)
53 \( 1 + (-9.74e4 - 1.12e5i)T^{2} \)
59 \( 1 + (1.97e5 - 5.78e4i)T^{2} \)
61 \( 1 + (-716. - 621. i)T + (3.23e4 + 2.24e5i)T^{2} \)
71 \( 1 + (2.34e5 + 2.70e5i)T^{2} \)
73 \( 1 + (695. - 802. i)T + (-5.53e4 - 3.85e5i)T^{2} \)
79 \( 1 + (712. + 102. i)T + (4.73e5 + 1.38e5i)T^{2} \)
83 \( 1 + (8.13e4 - 5.65e5i)T^{2} \)
89 \( 1 + (-5.93e5 - 3.81e5i)T^{2} \)
97 \( 1 + 1.69e3iT - 9.12e5T^{2} \)
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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

−11.88738490050748982061278316632, −11.45178339889164255105760529548, −10.00974292517149223766577261783, −8.549511716489384005116539612637, −8.068511994634904602973900622782, −6.85428769963375413734441453658, −5.89695388222628976609889613296, −4.25266624173478338186057264232, −2.93219920775896214281452297846, −1.01883740211686542060530639828, 0.942727064246679946033507193319, 3.27936782414273234388263243635, 4.58645669538112394750064559676, 5.62841512914436716631974916478, 6.42626936402316120110015518104, 8.379354712914974062192736491314, 9.228552871418533586855336044338, 10.03826542506853880310257817568, 10.97914892520177544064007957407, 11.62527238805180550595305196803

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