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Properties

Label 2-200-40.37-c2-0-31
Degree $2$
Conductor $200$
Sign $0.0146 + 0.999i$
Analytic cond. $5.44960$
Root an. cond. $2.33443$
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  + (1.96 − 0.376i)2-s + (−0.977 − 0.977i)3-s + (3.71 − 1.47i)4-s + (−2.28 − 1.55i)6-s + (−8.39 − 8.39i)7-s + (6.74 − 4.30i)8-s − 7.08i·9-s − 4.01i·11-s + (−5.07 − 2.18i)12-s + (11.0 + 11.0i)13-s + (−19.6 − 13.3i)14-s + (11.6 − 10.9i)16-s + (3.79 + 3.79i)17-s + (−2.66 − 13.9i)18-s + 15.9·19-s + ⋯
L(s)  = 1  + (0.982 − 0.188i)2-s + (−0.325 − 0.325i)3-s + (0.929 − 0.369i)4-s + (−0.381 − 0.258i)6-s + (−1.19 − 1.19i)7-s + (0.842 − 0.538i)8-s − 0.787i·9-s − 0.365i·11-s + (−0.423 − 0.182i)12-s + (0.852 + 0.852i)13-s + (−1.40 − 0.952i)14-s + (0.726 − 0.687i)16-s + (0.223 + 0.223i)17-s + (−0.148 − 0.773i)18-s + 0.840·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0146 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0146 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.0146 + 0.999i$
Analytic conductor: \(5.44960\)
Root analytic conductor: \(2.33443\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1),\ 0.0146 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60342 - 1.58013i\)
\(L(\frac12)\) \(\approx\) \(1.60342 - 1.58013i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 + 0.376i)T \)
5 \( 1 \)
good3 \( 1 + (0.977 + 0.977i)T + 9iT^{2} \)
7 \( 1 + (8.39 + 8.39i)T + 49iT^{2} \)
11 \( 1 + 4.01iT - 121T^{2} \)
13 \( 1 + (-11.0 - 11.0i)T + 169iT^{2} \)
17 \( 1 + (-3.79 - 3.79i)T + 289iT^{2} \)
19 \( 1 - 15.9T + 361T^{2} \)
23 \( 1 + (1.86 - 1.86i)T - 529iT^{2} \)
29 \( 1 + 0.468T + 841T^{2} \)
31 \( 1 + 17.3T + 961T^{2} \)
37 \( 1 + (22.1 - 22.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.0T + 1.68e3T^{2} \)
43 \( 1 + (-17.1 - 17.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-6.31 - 6.31i)T + 2.20e3iT^{2} \)
53 \( 1 + (-39.8 - 39.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 50.6T + 3.48e3T^{2} \)
61 \( 1 - 73.6iT - 3.72e3T^{2} \)
67 \( 1 + (-77.6 + 77.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 78.3T + 5.04e3T^{2} \)
73 \( 1 + (-46.0 + 46.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 31.6iT - 6.24e3T^{2} \)
83 \( 1 + (84.9 + 84.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 92.8iT - 7.92e3T^{2} \)
97 \( 1 + (-85.3 - 85.3i)T + 9.40e3iT^{2} \)
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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

−12.11910523863081886530088987011, −11.23901238102979946620928362279, −10.27598182604433381836990572320, −9.253226649716130837227309775237, −7.38514566638842549421031341374, −6.60048567212205516991864791966, −5.83057217118351989568467807126, −4.07631333565342415941619771208, −3.31072544310485400149178282749, −1.05906012288595868229846963541, 2.49303617059861916971012422613, 3.68648715270327445923328196564, 5.32038211178655526000940306363, 5.80772002248516037717702388183, 7.08487097113576399207252282100, 8.341052063475221082627374140138, 9.680153277505526422424735826109, 10.71456293969180147227232968097, 11.70213022357879612222819327493, 12.63230719293673876868972536880

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