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Properties

Label 2-200-40.29-c5-0-16
Degree $2$
Conductor $200$
Sign $0.658 - 0.752i$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.17 − 5.22i)2-s + 8.95·3-s + (−22.5 − 22.6i)4-s + (19.4 − 46.7i)6-s + 179. i·7-s + (−167. + 68.4i)8-s − 162.·9-s − 654. i·11-s + (−202. − 203. i)12-s + 160.·13-s + (935. + 389. i)14-s + (−6.40 + 1.02e3i)16-s + 347. i·17-s + (−353. + 850. i)18-s + 2.32e3i·19-s + ⋯
L(s)  = 1  + (0.384 − 0.923i)2-s + 0.574·3-s + (−0.704 − 0.709i)4-s + (0.220 − 0.530i)6-s + 1.38i·7-s + (−0.925 + 0.378i)8-s − 0.669·9-s − 1.62i·11-s + (−0.405 − 0.407i)12-s + 0.263·13-s + (1.27 + 0.530i)14-s + (−0.00625 + 0.999i)16-s + 0.292i·17-s + (−0.257 + 0.618i)18-s + 1.47i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.473328824\)
\(L(\frac12)\) \(\approx\) \(1.473328824\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.17 + 5.22i)T \)
5 \( 1 \)
good3 \( 1 - 8.95T + 243T^{2} \)
7 \( 1 - 179. iT - 1.68e4T^{2} \)
11 \( 1 + 654. iT - 1.61e5T^{2} \)
13 \( 1 - 160.T + 3.71e5T^{2} \)
17 \( 1 - 347. iT - 1.41e6T^{2} \)
19 \( 1 - 2.32e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.12e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.00e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.19e3T + 2.86e7T^{2} \)
37 \( 1 - 2.40e3T + 6.93e7T^{2} \)
41 \( 1 + 1.38e4T + 1.15e8T^{2} \)
43 \( 1 - 9.40e3T + 1.47e8T^{2} \)
47 \( 1 - 5.27e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 3.84e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.79e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.04e4T + 1.35e9T^{2} \)
71 \( 1 + 4.84e3T + 1.80e9T^{2} \)
73 \( 1 - 3.92e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.87e4T + 3.07e9T^{2} \)
83 \( 1 + 6.31e4T + 3.93e9T^{2} \)
89 \( 1 + 2.97e4T + 5.58e9T^{2} \)
97 \( 1 + 4.77e4iT - 8.58e9T^{2} \)
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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

−11.67484331968379377880473796786, −11.02482194651519136555365980813, −9.692955719658212865018953215109, −8.734303445444242065353856535190, −8.265627102713313542515492242107, −5.91244550953626197322045677822, −5.53807701252331792578426148013, −3.58572994657438425332100219426, −2.89745184446135079320714416359, −1.53763866440080489485356325107, 0.36053613734033053652662017920, 2.62744979170205141089951916659, 4.08698285756469822314816750885, 4.85345711992748127467056178803, 6.53660099432676596215727972787, 7.27368273068016498278294907770, 8.195369095956322736194076074024, 9.256738636253675854413134100869, 10.24520927303845603143880689310, 11.59559582714072449474982513072

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