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Properties

Label 2-201-201.194-c2-0-22
Degree $2$
Conductor $201$
Sign $0.968 - 0.247i$
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  + (2.87 + 0.845i)3-s + (2.32 − 3.25i)4-s + (−0.560 + 11.7i)7-s + (7.57 + 4.86i)9-s + (9.43 − 7.41i)12-s + (11.8 + 4.75i)13-s + (−5.23 − 15.1i)16-s + (−1.55 − 32.7i)19-s + (−11.5 + 33.3i)21-s + (−3.55 − 24.7i)25-s + (17.6 + 20.4i)27-s + (37.0 + 29.1i)28-s + (−34.6 + 13.8i)31-s + (33.4 − 13.3i)36-s + (−32.4 + 56.1i)37-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)3-s + (0.580 − 0.814i)4-s + (−0.0800 + 1.68i)7-s + (0.841 + 0.540i)9-s + (0.786 − 0.618i)12-s + (0.913 + 0.365i)13-s + (−0.327 − 0.945i)16-s + (−0.0820 − 1.72i)19-s + (−0.550 + 1.58i)21-s + (−0.142 − 0.989i)25-s + (0.654 + 0.755i)27-s + (1.32 + 1.03i)28-s + (−1.11 + 0.447i)31-s + (0.928 − 0.371i)36-s + (−0.876 + 1.51i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.968 - 0.247i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.33281 + 0.293629i\)
\(L(\frac12)\) \(\approx\) \(2.33281 + 0.293629i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.87 - 0.845i)T \)
67 \( 1 + (30.8 - 59.4i)T \)
good2 \( 1 + (-2.32 + 3.25i)T^{2} \)
5 \( 1 + (3.55 + 24.7i)T^{2} \)
7 \( 1 + (0.560 - 11.7i)T + (-48.7 - 4.65i)T^{2} \)
11 \( 1 + (-112. - 44.9i)T^{2} \)
13 \( 1 + (-11.8 - 4.75i)T + (122. + 116. i)T^{2} \)
17 \( 1 + (94.5 - 273. i)T^{2} \)
19 \( 1 + (1.55 + 32.7i)T + (-359. + 34.3i)T^{2} \)
23 \( 1 + (-25.1 + 528. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (34.6 - 13.8i)T + (695. - 663. i)T^{2} \)
37 \( 1 + (32.4 - 56.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-1.65e3 + 318. i)T^{2} \)
43 \( 1 + (-28.5 + 62.4i)T + (-1.21e3 - 1.39e3i)T^{2} \)
47 \( 1 + (1.96e3 - 1.01e3i)T^{2} \)
53 \( 1 + (1.83e3 - 2.12e3i)T^{2} \)
59 \( 1 + (3.33e3 + 980. i)T^{2} \)
61 \( 1 + (75.4 - 14.5i)T + (3.45e3 - 1.38e3i)T^{2} \)
71 \( 1 + (1.64e3 + 4.76e3i)T^{2} \)
73 \( 1 + (113. - 21.9i)T + (4.94e3 - 1.98e3i)T^{2} \)
79 \( 1 + (-83.3 + 65.5i)T + (1.47e3 - 6.06e3i)T^{2} \)
83 \( 1 + (5.41e3 - 4.25e3i)T^{2} \)
89 \( 1 + (-6.66e3 + 4.28e3i)T^{2} \)
97 \( 1 + (-66.2 + 114. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.18593186622698970201584358975, −11.23479821588937656823021995450, −10.21629671668754223134567078228, −9.047129936001625964525655117048, −8.689905931086949491800065316203, −7.08125447579306571491602458434, −5.98090892850158432772338509757, −4.82031736195853107531835613654, −2.97296439022538461235603547678, −1.93723293411883902598718991359, 1.58007225538203431605312158414, 3.44206758197312326612521681430, 3.95354146499585882055494628834, 6.26707157286055035371212530098, 7.51794270264040394100316822568, 7.75970222018742343583314179564, 9.043171278047047624751379213353, 10.32600544170124139462534786906, 11.09098431283006711250558395216, 12.48342129509224319043060978218

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