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Properties

Label 2-201-201.200-c1-0-15
Degree $2$
Conductor $201$
Sign $0.846 + 0.532i$
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  + 1.18·2-s + (1.24 − 1.19i)3-s − 0.588·4-s + 2.08·5-s + (1.48 − 1.42i)6-s − 0.284i·7-s − 3.07·8-s + (0.120 − 2.99i)9-s + 2.47·10-s − 1.80·11-s + (−0.734 + 0.705i)12-s + 5.82i·13-s − 0.337i·14-s + (2.60 − 2.50i)15-s − 2.47·16-s + 2.11i·17-s + ⋯
L(s)  = 1  + 0.840·2-s + (0.721 − 0.692i)3-s − 0.294·4-s + 0.932·5-s + (0.605 − 0.582i)6-s − 0.107i·7-s − 1.08·8-s + (0.0401 − 0.999i)9-s + 0.783·10-s − 0.543·11-s + (−0.212 + 0.203i)12-s + 1.61i·13-s − 0.0902i·14-s + (0.672 − 0.646i)15-s − 0.619·16-s + 0.513i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.846 + 0.532i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.846 + 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96195 - 0.566442i\)
\(L(\frac12)\) \(\approx\) \(1.96195 - 0.566442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.24 + 1.19i)T \)
67 \( 1 + (1.97 + 7.94i)T \)
good2 \( 1 - 1.18T + 2T^{2} \)
5 \( 1 - 2.08T + 5T^{2} \)
7 \( 1 + 0.284iT - 7T^{2} \)
11 \( 1 + 1.80T + 11T^{2} \)
13 \( 1 - 5.82iT - 13T^{2} \)
17 \( 1 - 2.11iT - 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 + 5.85iT - 23T^{2} \)
29 \( 1 + 1.18iT - 29T^{2} \)
31 \( 1 - 9.35iT - 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
41 \( 1 + 6.14T + 41T^{2} \)
43 \( 1 + 1.57iT - 43T^{2} \)
47 \( 1 - 3.59iT - 47T^{2} \)
53 \( 1 - 5.90T + 53T^{2} \)
59 \( 1 + 8.92iT - 59T^{2} \)
61 \( 1 - 3.69iT - 61T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 2.98T + 73T^{2} \)
79 \( 1 - 2.32iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 4.96iT - 89T^{2} \)
97 \( 1 - 8.79iT - 97T^{2} \)
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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

−12.57311426216131071582129511506, −11.92342014229576966084968253667, −10.27472612289611009686865471286, −9.208775992374659147596976831481, −8.539136705548517324224526975975, −6.99008980712532706744179048605, −6.10291625031799237616547074279, −4.82214601005108793683031126477, −3.44132212084251553803757098655, −2.00245856161857353542131417613, 2.64627075109504740437487492197, 3.68992886885158911851751740942, 5.27985521611972723862823134467, 5.57848024732554946444471838713, 7.57823480225749308722052807812, 8.714132903815299922378169302950, 9.704852173479648598148753511524, 10.27840066683757220637509427509, 11.71747859266998787548657295948, 13.08615207786291981155765400092

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