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Properties

Label 2-201-67.29-c3-0-9
Degree $2$
Conductor $201$
Sign $-0.638 - 0.769i$
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  + (−0.406 + 0.703i)2-s + 3·3-s + (3.66 + 6.35i)4-s − 6.19·5-s + (−1.21 + 2.11i)6-s + (−7.99 − 13.8i)7-s − 12.4·8-s + 9·9-s + (2.51 − 4.36i)10-s + (27.7 + 48.1i)11-s + (11.0 + 19.0i)12-s + (−39.8 + 69.0i)13-s + 12.9·14-s − 18.5·15-s + (−24.2 + 42.0i)16-s + (17.8 − 30.8i)17-s + ⋯
L(s)  = 1  + (−0.143 + 0.248i)2-s + 0.577·3-s + (0.458 + 0.794i)4-s − 0.554·5-s + (−0.0829 + 0.143i)6-s + (−0.431 − 0.747i)7-s − 0.551·8-s + 0.333·9-s + (0.0796 − 0.137i)10-s + (0.761 + 1.31i)11-s + (0.264 + 0.458i)12-s + (−0.850 + 1.47i)13-s + 0.248·14-s − 0.319·15-s + (−0.379 + 0.657i)16-s + (0.253 − 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.638 - 0.769i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.615003 + 1.30861i\)
\(L(\frac12)\) \(\approx\) \(0.615003 + 1.30861i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-531. - 136. i)T \)
good2 \( 1 + (0.406 - 0.703i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 6.19T + 125T^{2} \)
7 \( 1 + (7.99 + 13.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-27.7 - 48.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (39.8 - 69.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-17.8 + 30.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (21.3 - 37.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (1.49 - 2.59i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (41.2 + 71.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-120. - 209. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (104. - 180. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (7.40 + 12.8i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 178.T + 7.95e4T^{2} \)
47 \( 1 + (80.3 + 139. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 745.T + 1.48e5T^{2} \)
59 \( 1 + 30.4T + 2.05e5T^{2} \)
61 \( 1 + (24.8 - 43.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (-264. - 458. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-58.0 + 100. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-217. - 376. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-477. + 827. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 388.T + 7.04e5T^{2} \)
97 \( 1 + (-723. + 1.25e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.98440894973332258154337011716, −11.90321468441528032846408362091, −10.15530272769053916111689828504, −9.327349354866595607434824270456, −8.191117824061228310534659895753, −7.06134225274448572013030830886, −6.86698794179003245696913002833, −4.44646831169512300404557212338, −3.62925639319153618080883957095, −2.05225597768637708947562174229, 0.58398274094744525064083857426, 2.45976889386238427268613602011, 3.54646173763882824214352342394, 5.42728099078716008270341399067, 6.33283875686142504244101177120, 7.72052378816350670548093415176, 8.749477459860277018234998795585, 9.667138305019153521595277353780, 10.64181805752906173873418251895, 11.62191136624538632688608992416

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