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Properties

Label 2-201-67.30-c2-0-12
Degree $2$
Conductor $201$
Sign $0.475 + 0.879i$
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  + (−1.79 + 1.03i)2-s + 1.73i·3-s + (0.136 − 0.236i)4-s + 0.118i·5-s + (−1.79 − 3.10i)6-s + (−4.18 − 2.41i)7-s − 7.70i·8-s − 2.99·9-s + (−0.122 − 0.212i)10-s + (−7.76 − 4.48i)11-s + (0.409 + 0.236i)12-s + (3.99 − 2.30i)13-s + 9.99·14-s − 0.205·15-s + (8.50 + 14.7i)16-s + (−3.37 − 5.85i)17-s + ⋯
L(s)  = 1  + (−0.895 + 0.516i)2-s + 0.577i·3-s + (0.0341 − 0.0591i)4-s + 0.0237i·5-s + (−0.298 − 0.516i)6-s + (−0.598 − 0.345i)7-s − 0.962i·8-s − 0.333·9-s + (−0.0122 − 0.0212i)10-s + (−0.705 − 0.407i)11-s + (0.0341 + 0.0197i)12-s + (0.306 − 0.177i)13-s + 0.714·14-s − 0.0136·15-s + (0.531 + 0.921i)16-s + (−0.198 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.475 + 0.879i$
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: \((2,\ 201,\ (\ :1),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.317454 - 0.189188i\)
\(L(\frac12)\) \(\approx\) \(0.317454 - 0.189188i\)
\(L(2)\) 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 \( 1 + (63.0 + 22.6i)T \)
good2 \( 1 + (1.79 - 1.03i)T + (2 - 3.46i)T^{2} \)
5 \( 1 - 0.118iT - 25T^{2} \)
7 \( 1 + (4.18 + 2.41i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (7.76 + 4.48i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.99 + 2.30i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (3.37 + 5.85i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.77 - 4.80i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (18.0 + 31.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-14.9 + 25.8i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (0.770 + 0.445i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-3.63 - 6.28i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (58.6 + 33.8i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + 4.32iT - 1.84e3T^{2} \)
47 \( 1 + (-21.3 + 37.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 61.3iT - 2.80e3T^{2} \)
59 \( 1 + 45.4T + 3.48e3T^{2} \)
61 \( 1 + (48.7 - 28.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (38.7 - 67.1i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-17.3 - 30.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (81.4 + 47.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (54.8 + 94.9i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 5.63T + 7.92e3T^{2} \)
97 \( 1 + (-57.8 + 33.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−12.02108976318384193278360992714, −10.51871153147012470379457636503, −10.12190052277721244524463329464, −8.925272353506456112712445579286, −8.236254501808495205294305696799, −7.07723742756307647201519043989, −6.00414770330094166283913958559, −4.40629474419875502274745588301, −3.10059863254763672189735428255, −0.28647090074585190900119074314, 1.56256762817925900590897806166, 2.97391002001645395341310466688, 5.05660351853862378414681986569, 6.26172683081496308388364537297, 7.56092622022169890310348940946, 8.565116764641322082748269608010, 9.439602694600358864164282080781, 10.35206309873586884426641109042, 11.28430421357141081569475104465, 12.26433114631961423766047667171

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