Properties

Label 2-200-40.29-c5-0-22
Degree $2$
Conductor $200$
Sign $-0.559 - 0.828i$
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  + (3.03 + 4.77i)2-s − 23.6·3-s + (−13.5 + 28.9i)4-s + (−71.7 − 112. i)6-s − 160. i·7-s + (−179. + 23.4i)8-s + 314.·9-s + 129. i·11-s + (319. − 684. i)12-s + 759.·13-s + (766. − 488. i)14-s + (−657. − 785. i)16-s − 323. i·17-s + (955. + 1.50e3i)18-s + 198. i·19-s + ⋯
L(s)  = 1  + (0.537 + 0.843i)2-s − 1.51·3-s + (−0.423 + 0.906i)4-s + (−0.813 − 1.27i)6-s − 1.23i·7-s + (−0.991 + 0.129i)8-s + 1.29·9-s + 0.321i·11-s + (0.641 − 1.37i)12-s + 1.24·13-s + (1.04 − 0.665i)14-s + (−0.641 − 0.766i)16-s − 0.271i·17-s + (0.694 + 1.09i)18-s + 0.126i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.559 - 0.828i$
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: \((2,\ 200,\ (\ :5/2),\ -0.559 - 0.828i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.057610142\)
\(L(\frac12)\) \(\approx\) \(1.057610142\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.03 - 4.77i)T \)
5 \( 1 \)
good3 \( 1 + 23.6T + 243T^{2} \)
7 \( 1 + 160. iT - 1.68e4T^{2} \)
11 \( 1 - 129. iT - 1.61e5T^{2} \)
13 \( 1 - 759.T + 3.71e5T^{2} \)
17 \( 1 + 323. iT - 1.41e6T^{2} \)
19 \( 1 - 198. iT - 2.47e6T^{2} \)
23 \( 1 + 1.19e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.98e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.87e3T + 2.86e7T^{2} \)
37 \( 1 - 3.69e3T + 6.93e7T^{2} \)
41 \( 1 + 1.04e4T + 1.15e8T^{2} \)
43 \( 1 + 9.87e3T + 1.47e8T^{2} \)
47 \( 1 - 6.29e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.17e4T + 4.18e8T^{2} \)
59 \( 1 - 3.35e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.85e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.31e4T + 1.35e9T^{2} \)
71 \( 1 - 5.94e4T + 1.80e9T^{2} \)
73 \( 1 - 5.12e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.37e4T + 3.07e9T^{2} \)
83 \( 1 + 6.16e4T + 3.93e9T^{2} \)
89 \( 1 - 1.06e5T + 5.58e9T^{2} \)
97 \( 1 + 1.25e4iT - 8.58e9T^{2} \)
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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

−11.97247456744943870407378387559, −11.05119202230623792000650351620, −10.25718524090950503818541845904, −8.724731470689100747741009098118, −7.34269566585559942525691747840, −6.66641943925685524549875271589, −5.69373039698848417842748634514, −4.68315448563793631038728434550, −3.67671678225441656285880280395, −0.944422532740760149998556910518, 0.44022497863143795405037164598, 1.86975325794326641026882816847, 3.56089671973284099074275978843, 4.99389689761302644042274935914, 5.80837380629824899490847258731, 6.40771621911847821791446893386, 8.478920549827805365274689143824, 9.586260465007933732388494846752, 10.72135896235184130891089468380, 11.41637206338824983896004801419

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