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Properties

Label 2-201-201.200-c1-0-12
Degree $2$
Conductor $201$
Sign $-0.557 + 0.830i$
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  − 2.09·2-s + (1.16 − 1.28i)3-s + 2.38·4-s − 1.47·5-s + (−2.44 + 2.68i)6-s − 2.11i·7-s − 0.798·8-s + (−0.281 − 2.98i)9-s + 3.09·10-s − 1.98·11-s + (2.77 − 3.05i)12-s + 0.757i·13-s + 4.43i·14-s + (−1.72 + 1.89i)15-s − 3.09·16-s − 4.64i·17-s + ⋯
L(s)  = 1  − 1.48·2-s + (0.673 − 0.739i)3-s + 1.19·4-s − 0.660·5-s + (−0.996 + 1.09i)6-s − 0.800i·7-s − 0.282·8-s + (−0.0936 − 0.995i)9-s + 0.977·10-s − 0.598·11-s + (0.801 − 0.880i)12-s + 0.210i·13-s + 1.18i·14-s + (−0.444 + 0.488i)15-s − 0.772·16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.557 + 0.830i$
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: \((2,\ 201,\ (\ :1/2),\ -0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240982 - 0.451980i\)
\(L(\frac12)\) \(\approx\) \(0.240982 - 0.451980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.16 + 1.28i)T \)
67 \( 1 + (-8.09 + 1.20i)T \)
good2 \( 1 + 2.09T + 2T^{2} \)
5 \( 1 + 1.47T + 5T^{2} \)
7 \( 1 + 2.11iT - 7T^{2} \)
11 \( 1 + 1.98T + 11T^{2} \)
13 \( 1 - 0.757iT - 13T^{2} \)
17 \( 1 + 4.64iT - 17T^{2} \)
19 \( 1 + 1.49T + 19T^{2} \)
23 \( 1 + 3.56iT - 23T^{2} \)
29 \( 1 + 2.91iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 3.85T + 37T^{2} \)
41 \( 1 - 3.03T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 5.79iT - 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 14.2iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
71 \( 1 + 4.38iT - 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 17.6iT - 79T^{2} \)
83 \( 1 + 5.31iT - 83T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 - 7.55iT - 97T^{2} \)
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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

−11.85835496683223497813582738013, −10.99247346992208292679814096809, −9.882889847387448772357308575470, −9.051330913100622476178725064120, −7.85222501401140291349079621177, −7.64355739158382826725692599168, −6.51328783081395171341361164670, −4.25106205411874683721769595863, −2.48920001476173565348504260440, −0.65690523542256897417039213024, 2.22954449860321325758363458902, 3.83781415917556335252691719729, 5.42214446095635587080026995029, 7.23119601392635095528912307162, 8.279133905871214250979244306377, 8.642072613117398653234514601772, 9.740256616399277493015992931741, 10.52403251837990720045509584174, 11.34763389815165957886985905381, 12.60589502840513581665486813594

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