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Properties

Label 2-201-201.38-c1-0-14
Degree $2$
Conductor $201$
Sign $0.873 + 0.486i$
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  + (0.834 − 1.44i)2-s + (1.24 + 1.20i)3-s + (−0.393 − 0.681i)4-s + 0.655·5-s + (2.78 − 0.787i)6-s + (−1.66 + 0.960i)7-s + 2.02·8-s + (0.0839 + 2.99i)9-s + (0.546 − 0.947i)10-s + (−1.39 − 2.41i)11-s + (0.334 − 1.32i)12-s + (−3.35 − 1.93i)13-s + 3.20i·14-s + (0.813 + 0.791i)15-s + (2.47 − 4.29i)16-s + (−0.511 − 0.295i)17-s + ⋯
L(s)  = 1  + (0.590 − 1.02i)2-s + (0.716 + 0.697i)3-s + (−0.196 − 0.340i)4-s + 0.292·5-s + (1.13 − 0.321i)6-s + (−0.628 + 0.362i)7-s + 0.716·8-s + (0.0279 + 0.999i)9-s + (0.172 − 0.299i)10-s + (−0.420 − 0.728i)11-s + (0.0964 − 0.381i)12-s + (−0.930 − 0.537i)13-s + 0.856i·14-s + (0.210 + 0.204i)15-s + (0.619 − 1.07i)16-s + (−0.124 − 0.0716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.873 + 0.486i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88676 - 0.489398i\)
\(L(\frac12)\) \(\approx\) \(1.88676 - 0.489398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.24 - 1.20i)T \)
67 \( 1 + (3.68 - 7.30i)T \)
good2 \( 1 + (-0.834 + 1.44i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.655T + 5T^{2} \)
7 \( 1 + (1.66 - 0.960i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.39 + 2.41i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.35 + 1.93i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.511 + 0.295i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.700 + 1.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 + 0.714i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.80 + 3.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 1.47i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.19 - 2.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.53 + 4.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + (-2.53 + 1.46i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 8.85iT - 59T^{2} \)
61 \( 1 + (-6.39 - 3.69i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (2.63 - 1.52i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.28 + 9.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.66 + 3.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.63 + 5.56i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.84iT - 89T^{2} \)
97 \( 1 + (-5.96 - 3.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.42311528552636183015387747593, −11.43053841340299043985615353483, −10.32399164244103380281322435793, −9.828946873667660144283202059125, −8.585338577574221536383351731720, −7.46469735509011368630895581689, −5.65824870732821449536535377048, −4.49154757873025587440617268646, −3.19921467635483226089498518362, −2.43774131766616202102697302966, 2.10818367805251512847581068802, 3.90696613135605112117840639157, 5.31106417429556323654204933883, 6.60391900386405223182613633900, 7.16617569156240370622758889493, 8.103792504088171605761944925167, 9.498701811946500496960531233615, 10.30445172739956943021558682248, 12.02471439515229273141863227580, 12.88579027392027947280209588005

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