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Properties

Label 2-200-40.29-c5-0-18
Degree $2$
Conductor $200$
Sign $-0.877 - 0.479i$
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  + (5.36 + 1.78i)2-s − 29.6·3-s + (25.6 + 19.1i)4-s + (−158. − 52.8i)6-s + 90.3i·7-s + (103. + 148. i)8-s + 633.·9-s − 181. i·11-s + (−758. − 567. i)12-s + 455.·13-s + (−161. + 484. i)14-s + (289. + 982. i)16-s + 615. i·17-s + (3.39e3 + 1.13e3i)18-s − 2.49e3i·19-s + ⋯
L(s)  = 1  + (0.948 + 0.315i)2-s − 1.89·3-s + (0.800 + 0.598i)4-s + (−1.80 − 0.599i)6-s + 0.696i·7-s + (0.570 + 0.821i)8-s + 2.60·9-s − 0.453i·11-s + (−1.52 − 1.13i)12-s + 0.747·13-s + (−0.219 + 0.661i)14-s + (0.282 + 0.959i)16-s + 0.516i·17-s + (2.47 + 0.822i)18-s − 1.58i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.877 - 0.479i$
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: \((2,\ 200,\ (\ :5/2),\ -0.877 - 0.479i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.332888518\)
\(L(\frac12)\) \(\approx\) \(1.332888518\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.36 - 1.78i)T \)
5 \( 1 \)
good3 \( 1 + 29.6T + 243T^{2} \)
7 \( 1 - 90.3iT - 1.68e4T^{2} \)
11 \( 1 + 181. iT - 1.61e5T^{2} \)
13 \( 1 - 455.T + 3.71e5T^{2} \)
17 \( 1 - 615. iT - 1.41e6T^{2} \)
19 \( 1 + 2.49e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.42e3iT - 6.43e6T^{2} \)
29 \( 1 - 7.65e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.76e3T + 2.86e7T^{2} \)
37 \( 1 + 4.80e3T + 6.93e7T^{2} \)
41 \( 1 - 1.88e3T + 1.15e8T^{2} \)
43 \( 1 + 8.26e3T + 1.47e8T^{2} \)
47 \( 1 + 5.72e3iT - 2.29e8T^{2} \)
53 \( 1 + 3.23e4T + 4.18e8T^{2} \)
59 \( 1 - 2.08e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.32e4iT - 8.44e8T^{2} \)
67 \( 1 + 9.64e3T + 1.35e9T^{2} \)
71 \( 1 + 3.30e4T + 1.80e9T^{2} \)
73 \( 1 - 1.85e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.42e4T + 3.07e9T^{2} \)
83 \( 1 - 8.00e4T + 3.93e9T^{2} \)
89 \( 1 - 8.16e4T + 5.58e9T^{2} \)
97 \( 1 - 2.22e4iT - 8.58e9T^{2} \)
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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

−11.94615182785812247934268978459, −11.22498157251420753317502210096, −10.69620110348114339947433981451, −9.018131780419319780499621955095, −7.36108455152319737511471437104, −6.46732172985770850216515713694, −5.58571214072071075186158480595, −4.99725383149491547980106575310, −3.56774076869559653260006098686, −1.49849464332484990613840667508, 0.39051672344127429680413428627, 1.63645452768076421994459882331, 3.91151340962259022176098959313, 4.72399931102422651688658593420, 5.85848956651450668685877317303, 6.52427710170716192406090813697, 7.56873568567050111715459406652, 9.908940347575238325220634060207, 10.54867834166984366754093129325, 11.27234854494316644256295808929

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