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Properties

Label 2-200-8.3-c2-0-3
Degree $2$
Conductor $200$
Sign $0.238 - 0.971i$
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  + (−1.64 − 1.13i)2-s − 2.54·3-s + (1.41 + 3.73i)4-s + (4.19 + 2.89i)6-s − 8.05i·7-s + (1.90 − 7.76i)8-s − 2.51·9-s + 0.942·11-s + (−3.61 − 9.52i)12-s + 18.4i·13-s + (−9.15 + 13.2i)14-s + (−11.9 + 10.6i)16-s − 19.8·17-s + (4.13 + 2.85i)18-s + 20.8·19-s + ⋯
L(s)  = 1  + (−0.823 − 0.567i)2-s − 0.849·3-s + (0.354 + 0.934i)4-s + (0.698 + 0.482i)6-s − 1.15i·7-s + (0.238 − 0.971i)8-s − 0.278·9-s + 0.0856·11-s + (−0.301 − 0.793i)12-s + 1.42i·13-s + (−0.653 + 0.947i)14-s + (−0.747 + 0.663i)16-s − 1.16·17-s + (0.229 + 0.158i)18-s + 1.09·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(5.44960\)
Root analytic conductor: \(2.33443\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1),\ 0.238 - 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.294575 + 0.230931i\)
\(L(\frac12)\) \(\approx\) \(0.294575 + 0.230931i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.64 + 1.13i)T \)
5 \( 1 \)
good3 \( 1 + 2.54T + 9T^{2} \)
7 \( 1 + 8.05iT - 49T^{2} \)
11 \( 1 - 0.942T + 121T^{2} \)
13 \( 1 - 18.4iT - 169T^{2} \)
17 \( 1 + 19.8T + 289T^{2} \)
19 \( 1 - 20.8T + 361T^{2} \)
23 \( 1 - 39.0iT - 529T^{2} \)
29 \( 1 - 13.7iT - 841T^{2} \)
31 \( 1 - 18.7iT - 961T^{2} \)
37 \( 1 - 52.7iT - 1.36e3T^{2} \)
41 \( 1 + 69.0T + 1.68e3T^{2} \)
43 \( 1 + 34.3T + 1.84e3T^{2} \)
47 \( 1 - 10.7iT - 2.20e3T^{2} \)
53 \( 1 + 29.5iT - 2.80e3T^{2} \)
59 \( 1 - 7.72T + 3.48e3T^{2} \)
61 \( 1 - 89.1iT - 3.72e3T^{2} \)
67 \( 1 - 2.10T + 4.48e3T^{2} \)
71 \( 1 + 115. iT - 5.04e3T^{2} \)
73 \( 1 - 88.0T + 5.32e3T^{2} \)
79 \( 1 + 28.2iT - 6.24e3T^{2} \)
83 \( 1 + 122.T + 6.88e3T^{2} \)
89 \( 1 + 44.5T + 7.92e3T^{2} \)
97 \( 1 - 97.8T + 9.40e3T^{2} \)
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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

−11.77291452445700543668972548961, −11.56253645616375842896658889601, −10.58324327369463073801692105595, −9.646987192717424263985817111381, −8.626312476788406807438907202737, −7.25254189522226178430631666625, −6.58409629320157835818932131334, −4.81613793143037782524770574019, −3.47231452725464026638595534190, −1.42903089204889923610849149457, 0.32184722226063616459820280488, 2.53405347699606868515124209476, 5.08938532250168073484209178790, 5.79586576551355919291078158204, 6.72710685780603786027415261459, 8.153437839148731907005911675718, 8.874352363872754096831861654591, 10.05203871513710161884265326338, 10.99466147835650360485252738009, 11.76558806192599677137636172260

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