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Properties

Label 2-201-201.191-c1-0-14
Degree $2$
Conductor $201$
Sign $0.716 + 0.697i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.71 − 0.246i)3-s + (−0.0951 − 1.99i)4-s + (−0.730 − 0.177i)7-s + (2.87 − 0.845i)9-s + (−0.655 − 3.40i)12-s + (−0.293 − 0.101i)13-s + (−3.98 + 0.380i)16-s + (1.07 + 4.43i)19-s + (−1.29 − 0.123i)21-s + (3.27 + 3.77i)25-s + (4.72 − 2.15i)27-s + (−0.284 + 1.47i)28-s + (−2.61 + 0.905i)31-s + (−1.96 − 5.67i)36-s + (0.656 + 1.13i)37-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)3-s + (−0.0475 − 0.998i)4-s + (−0.276 − 0.0669i)7-s + (0.959 − 0.281i)9-s + (−0.189 − 0.981i)12-s + (−0.0813 − 0.0281i)13-s + (−0.995 + 0.0950i)16-s + (0.247 + 1.01i)19-s + (−0.282 − 0.0270i)21-s + (0.654 + 0.755i)25-s + (0.909 − 0.415i)27-s + (−0.0537 + 0.278i)28-s + (−0.470 + 0.162i)31-s + (−0.327 − 0.945i)36-s + (0.107 + 0.186i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43324 - 0.582187i\)
\(L(\frac12)\) \(\approx\) \(1.43324 - 0.582187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.246i)T \)
67 \( 1 + (-7.52 + 3.21i)T \)
good2 \( 1 + (0.0951 + 1.99i)T^{2} \)
5 \( 1 + (-3.27 - 3.77i)T^{2} \)
7 \( 1 + (0.730 + 0.177i)T + (6.22 + 3.20i)T^{2} \)
11 \( 1 + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.293 + 0.101i)T + (10.2 + 8.03i)T^{2} \)
17 \( 1 + (16.9 + 1.61i)T^{2} \)
19 \( 1 + (-1.07 - 4.43i)T + (-16.8 + 8.70i)T^{2} \)
23 \( 1 + (-5.42 + 22.3i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.61 - 0.905i)T + (24.3 - 19.1i)T^{2} \)
37 \( 1 + (-0.656 - 1.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (23.7 - 33.3i)T^{2} \)
43 \( 1 + (6.33 - 9.85i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-34.0 - 32.4i)T^{2} \)
53 \( 1 + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (9.47 + 6.74i)T + (19.9 + 57.6i)T^{2} \)
71 \( 1 + (70.6 - 6.74i)T^{2} \)
73 \( 1 + (5.01 - 7.03i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (1.78 + 9.27i)T + (-73.3 + 29.3i)T^{2} \)
83 \( 1 + (-81.5 + 15.7i)T^{2} \)
89 \( 1 + (85.3 + 25.0i)T^{2} \)
97 \( 1 + (-16.9 + 9.77i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.57720031134922372026873659118, −11.22082981333663424216628705723, −10.09672203995669882338721195446, −9.487767185127571090233267772708, −8.432711271895739910372493619909, −7.24836513877817262106395249955, −6.15427308142576355149875813356, −4.77324286948296951687032223200, −3.28393043902929117998432012741, −1.62312422750657091741948922819, 2.50094566232790761025452736648, 3.57630554305632695469128406861, 4.76779452881453666921210760966, 6.73974594170977498735602072989, 7.63116096771281802493949599563, 8.637065336595865945672660294997, 9.316290526568746800950562112058, 10.52116805806710781379593304931, 11.77800284355052972874475008276, 12.76306334934882295662471953595

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