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Properties

Label 2-200-40.29-c1-0-9
Degree $2$
Conductor $200$
Sign $0.813 + 0.581i$
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  + (−1.32 − 0.5i)2-s + 2.64·3-s + (1.50 + 1.32i)4-s + (−3.50 − 1.32i)6-s − 4i·7-s + (−1.32 − 2.50i)8-s + 4.00·9-s + 2.64i·11-s + (3.96 + 3.50i)12-s + (−2 + 5.29i)14-s + (0.500 + 3.96i)16-s + 3i·17-s + (−5.29 − 2.00i)18-s − 2.64i·19-s − 10.5i·21-s + (1.32 − 3.50i)22-s + ⋯
L(s)  = 1  + (−0.935 − 0.353i)2-s + 1.52·3-s + (0.750 + 0.661i)4-s + (−1.42 − 0.540i)6-s − 1.51i·7-s + (−0.467 − 0.883i)8-s + 1.33·9-s + 0.797i·11-s + (1.14 + 1.01i)12-s + (−0.534 + 1.41i)14-s + (0.125 + 0.992i)16-s + 0.727i·17-s + (−1.24 − 0.471i)18-s − 0.606i·19-s − 2.30i·21-s + (0.282 − 0.746i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1/2),\ 0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19519 - 0.383154i\)
\(L(\frac12)\) \(\approx\) \(1.19519 - 0.383154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.5i)T \)
5 \( 1 \)
good3 \( 1 - 2.64T + 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 2.64iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 5.29iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 - 7.93T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 7iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−12.46221189743917070243469084158, −11.03243524296027662884972981176, −10.12767784143628894339140813921, −9.439018214295961896902683883830, −8.375610926513820072529665094237, −7.56656523134970228466926862886, −6.85313421774544453589408028956, −4.21054357404571717333386696193, −3.19503564901073142508403368303, −1.67095650089860251827778479311, 2.13070617041045605093388771132, 3.14325443589533872306967242419, 5.37092937476263134399023166095, 6.63224920823709166006098030981, 8.001034426001162320652754232316, 8.605537402077930067745739153173, 9.190607393385172179734590056136, 10.16760861215149777612114999159, 11.51348187918869411358837510385, 12.49464969659128353334291672713

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