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Properties

Label 2-200-40.29-c5-0-0
Degree $2$
Conductor $200$
Sign $-0.423 + 0.905i$
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  + (−5.65 + 0.0494i)2-s − 10.7·3-s + (31.9 − 0.559i)4-s + (60.7 − 0.531i)6-s + 198. i·7-s + (−180. + 4.74i)8-s − 127.·9-s − 85.9i·11-s + (−343. + 6.01i)12-s − 407.·13-s + (−9.83 − 1.12e3i)14-s + (1.02e3 − 35.8i)16-s + 1.20e3i·17-s + (721. − 6.31i)18-s − 206. i·19-s + ⋯
L(s)  = 1  + (−0.999 + 0.00874i)2-s − 0.689·3-s + (0.999 − 0.0174i)4-s + (0.689 − 0.00602i)6-s + 1.53i·7-s + (−0.999 + 0.0262i)8-s − 0.524·9-s − 0.214i·11-s + (−0.689 + 0.0120i)12-s − 0.668·13-s + (−0.0134 − 1.53i)14-s + (0.999 − 0.0349i)16-s + 1.01i·17-s + (0.524 − 0.00459i)18-s − 0.130i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.423 + 0.905i$
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: \((2,\ 200,\ (\ :5/2),\ -0.423 + 0.905i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.003213005511\)
\(L(\frac12)\) \(\approx\) \(0.003213005511\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.65 - 0.0494i)T \)
5 \( 1 \)
good3 \( 1 + 10.7T + 243T^{2} \)
7 \( 1 - 198. iT - 1.68e4T^{2} \)
11 \( 1 + 85.9iT - 1.61e5T^{2} \)
13 \( 1 + 407.T + 3.71e5T^{2} \)
17 \( 1 - 1.20e3iT - 1.41e6T^{2} \)
19 \( 1 + 206. iT - 2.47e6T^{2} \)
23 \( 1 - 2.59e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.19e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.47e4T + 6.93e7T^{2} \)
41 \( 1 - 1.80e4T + 1.15e8T^{2} \)
43 \( 1 - 9.26e3T + 1.47e8T^{2} \)
47 \( 1 + 2.43e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.27e4T + 4.18e8T^{2} \)
59 \( 1 + 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.13e4iT - 8.44e8T^{2} \)
67 \( 1 + 6.26e4T + 1.35e9T^{2} \)
71 \( 1 + 6.12e4T + 1.80e9T^{2} \)
73 \( 1 + 2.32e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.91e4T + 3.07e9T^{2} \)
83 \( 1 + 4.80e4T + 3.93e9T^{2} \)
89 \( 1 + 3.01e4T + 5.58e9T^{2} \)
97 \( 1 + 1.13e5iT - 8.58e9T^{2} \)
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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

−12.02344239937908815349989043219, −11.23706496586963041520551773289, −10.32767269529565846222888466624, −9.070358036025824218034371227794, −8.560253576365945806985589843648, −7.21967566435312411039286665449, −5.96179922237531696913522603446, −5.39049944890737178921310889740, −3.06650079434257865881185657630, −1.80233867916748017587575934618, 0.00190138533324276936429834245, 0.897585502758374038787040452316, 2.69361110111924550426142252481, 4.41104445710747377246687693735, 5.88908594769223287747729278596, 7.03302459477569725715302724757, 7.68250591295375719892279220626, 9.018060798935849029731662591447, 10.11745618765361612426235191486, 10.74276119919313163811167999091

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