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Properties

Label 2-200-25.11-c1-0-5
Degree $2$
Conductor $200$
Sign $-0.187 + 0.982i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  + (0.690 − 2.12i)3-s + (−1.80 − 1.31i)5-s + 0.381·7-s + (−1.61 − 1.17i)9-s + (2.61 − 1.90i)11-s + (−4.54 − 3.30i)13-s + (−4.04 + 2.93i)15-s + (0.236 + 0.726i)17-s + (1.66 + 5.11i)19-s + (0.263 − 0.812i)21-s + (2.80 − 2.04i)23-s + (1.54 + 4.75i)25-s + (1.80 − 1.31i)27-s + (2.04 − 6.29i)29-s + (2.69 + 8.28i)31-s + ⋯
L(s)  = 1  + (0.398 − 1.22i)3-s + (−0.809 − 0.587i)5-s + 0.144·7-s + (−0.539 − 0.391i)9-s + (0.789 − 0.573i)11-s + (−1.26 − 0.915i)13-s + (−1.04 + 0.758i)15-s + (0.0572 + 0.176i)17-s + (0.381 + 1.17i)19-s + (0.0575 − 0.177i)21-s + (0.585 − 0.425i)23-s + (0.309 + 0.951i)25-s + (0.348 − 0.252i)27-s + (0.379 − 1.16i)29-s + (0.483 + 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ -0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753939 - 0.911356i\)
\(L(\frac12)\) \(\approx\) \(0.753939 - 0.911356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.80 + 1.31i)T \)
good3 \( 1 + (-0.690 + 2.12i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 0.381T + 7T^{2} \)
11 \( 1 + (-2.61 + 1.90i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.54 + 3.30i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.236 - 0.726i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.66 - 5.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.80 + 2.04i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-2.04 + 6.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.69 - 8.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.04 - 2.21i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-6.23 - 4.53i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.61T + 43T^{2} \)
47 \( 1 + (-2.11 + 6.51i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.07 - 9.45i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.73 + 2.71i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (7.42 - 5.39i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.0901 - 0.277i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.97 - 6.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.89 - 6.46i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.19 + 9.82i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.16 + 12.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.23 - 2.35i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.64 + 17.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.32948248778445937128870309131, −11.63736653099691810447406156225, −10.22383703559096009277236116306, −8.859993352833541823017662914576, −7.963095905909545202809939208296, −7.38688724523140684504483379588, −6.07248932314260346534353139734, −4.57583726517368218232392400119, −2.97578655875082507004543226993, −1.12208157242451914798598511815, 2.79818339854156340071020354116, 4.13023400979335061773916604682, 4.81081477263425944096717602594, 6.77508143638941654374302885484, 7.62136861894541484636262768885, 9.192739938592850615959794040886, 9.548089632907229589582055311377, 10.81694224337134835432520764626, 11.55500109148100400191305351101, 12.52099533652145591866653048902

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