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Properties

Label 2-201-67.30-c2-0-5
Degree $2$
Conductor $201$
Sign $0.476 - 0.879i$
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  + (2.13 − 1.23i)2-s + 1.73i·3-s + (1.02 − 1.77i)4-s + 7.82i·5-s + (2.13 + 3.69i)6-s + (−7.43 − 4.29i)7-s + 4.78i·8-s − 2.99·9-s + (9.62 + 16.6i)10-s + (11.6 + 6.70i)11-s + (3.08 + 1.77i)12-s + (4.70 − 2.71i)13-s − 21.1·14-s − 13.5·15-s + (9.99 + 17.3i)16-s + (12.6 + 21.9i)17-s + ⋯
L(s)  = 1  + (1.06 − 0.615i)2-s + 0.577i·3-s + (0.256 − 0.444i)4-s + 1.56i·5-s + (0.355 + 0.615i)6-s + (−1.06 − 0.613i)7-s + 0.598i·8-s − 0.333·9-s + (0.962 + 1.66i)10-s + (1.05 + 0.609i)11-s + (0.256 + 0.148i)12-s + (0.361 − 0.208i)13-s − 1.50·14-s − 0.903·15-s + (0.624 + 1.08i)16-s + (0.746 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.476 - 0.879i$
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: \((2,\ 201,\ (\ :1),\ 0.476 - 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.97080 + 1.17354i\)
\(L(\frac12)\) \(\approx\) \(1.97080 + 1.17354i\)
\(L(2)\) 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 \( 1 + (-15.5 - 65.1i)T \)
good2 \( 1 + (-2.13 + 1.23i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 7.82iT - 25T^{2} \)
7 \( 1 + (7.43 + 4.29i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-11.6 - 6.70i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.70 + 2.71i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-12.6 - 21.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (15.3 + 26.6i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (8.56 + 14.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-21.0 + 36.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-46.4 - 26.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-23.6 - 40.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (17.4 + 10.0i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + 80.1iT - 1.84e3T^{2} \)
47 \( 1 + (-3.01 + 5.22i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 1.34iT - 2.80e3T^{2} \)
59 \( 1 + 37.0T + 3.48e3T^{2} \)
61 \( 1 + (-3.90 + 2.25i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (6.61 - 11.4i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-45.0 - 78.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (124. + 71.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-69.1 - 119. i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 83.5T + 7.92e3T^{2} \)
97 \( 1 + (-49.0 + 28.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.35421542959290466352687492000, −11.47053803675025155175949852962, −10.42472523288187791077892361127, −10.08486461791293163308366757528, −8.446937936022498246003549603316, −6.77442392314765679117992474731, −6.19610398792625716114797003750, −4.35070053956121185291494235683, −3.58960815311850220041056906238, −2.62725578275889276307284267613, 1.02755315717529875001852664362, 3.41879374763720287752233585028, 4.67900468462456077253953917270, 5.89534570973707601194355217451, 6.34349204679572575638553010904, 7.88390056867941770297146244972, 9.038469295144671815779023720697, 9.717622487163391456565474282140, 11.82396688240941114979479234855, 12.34221517684033799151144298791

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