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Properties

Label 2-201-67.37-c3-0-9
Degree $2$
Conductor $201$
Sign $-0.545 - 0.837i$
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  + (1.18 + 2.05i)2-s + 3·3-s + (1.19 − 2.06i)4-s − 8.96·5-s + (3.55 + 6.15i)6-s + (−14.9 + 25.8i)7-s + 24.6·8-s + 9·9-s + (−10.6 − 18.3i)10-s + (−20.9 + 36.2i)11-s + (3.57 − 6.18i)12-s + (7.86 + 13.6i)13-s − 70.7·14-s − 26.8·15-s + (19.6 + 34.0i)16-s + (17.4 + 30.2i)17-s + ⋯
L(s)  = 1  + (0.419 + 0.725i)2-s + 0.577·3-s + (0.148 − 0.257i)4-s − 0.801·5-s + (0.241 + 0.419i)6-s + (−0.805 + 1.39i)7-s + 1.08·8-s + 0.333·9-s + (−0.335 − 0.581i)10-s + (−0.573 + 0.993i)11-s + (0.0859 − 0.148i)12-s + (0.167 + 0.290i)13-s − 1.34·14-s − 0.462·15-s + (0.306 + 0.531i)16-s + (0.249 + 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.545 - 0.837i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.545 - 0.837i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.988025 + 1.82241i\)
\(L(\frac12)\) \(\approx\) \(0.988025 + 1.82241i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 + (27.3 - 547. i)T \)
good2 \( 1 + (-1.18 - 2.05i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 8.96T + 125T^{2} \)
7 \( 1 + (14.9 - 25.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (20.9 - 36.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-7.86 - 13.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-17.4 - 30.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-25.6 - 44.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-76.1 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-61.3 + 106. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (4.58 - 7.94i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (65.2 + 112. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-152. + 263. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 347.T + 7.95e4T^{2} \)
47 \( 1 + (112. - 194. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 37.8T + 1.48e5T^{2} \)
59 \( 1 - 588.T + 2.05e5T^{2} \)
61 \( 1 + (398. + 690. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-156. + 271. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-179. - 310. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-274. + 475. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (124. + 214. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.39e3T + 7.04e5T^{2} \)
97 \( 1 + (-281. - 486. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.48102119189876210304269732065, −11.57522199541185115288678514490, −10.17421683735056113698379848623, −9.303848606810305544143986327599, −8.063704568979080147456163830825, −7.23951109585967814729450620649, −6.07908546456751189774597651488, −5.04676345237224326497634887384, −3.61016699172960573641988463896, −2.07088765446637159508941584672, 0.73200474701477890118176372227, 2.96591228425568851983164706747, 3.55159539653807162097884124896, 4.68376253122183279248743338223, 6.77329732760999927325497611536, 7.61150094235262374040976182742, 8.496465947220765757096014195113, 10.05459734422802890360172469095, 10.78371220131908546349250458870, 11.63438056541390339342157895946

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