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Properties

Label 2-200-40.29-c5-0-21
Degree $2$
Conductor $200$
Sign $-0.392 - 0.919i$
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  + (−4.95 + 2.73i)2-s − 18.7·3-s + (17.0 − 27.0i)4-s + (92.6 − 51.0i)6-s + 207. i·7-s + (−10.8 + 180. i)8-s + 106.·9-s − 248. i·11-s + (−319. + 506. i)12-s + 532.·13-s + (−567. − 1.03e3i)14-s + (−439. − 924. i)16-s − 710. i·17-s + (−529. + 291. i)18-s + 1.34e3i·19-s + ⋯
L(s)  = 1  + (−0.875 + 0.482i)2-s − 1.20·3-s + (0.534 − 0.845i)4-s + (1.05 − 0.579i)6-s + 1.60i·7-s + (−0.0597 + 0.998i)8-s + 0.440·9-s − 0.618i·11-s + (−0.640 + 1.01i)12-s + 0.874·13-s + (−0.774 − 1.40i)14-s + (−0.429 − 0.903i)16-s − 0.596i·17-s + (−0.385 + 0.212i)18-s + 0.856i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.392 - 0.919i$
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: \((2,\ 200,\ (\ :5/2),\ -0.392 - 0.919i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6320447681\)
\(L(\frac12)\) \(\approx\) \(0.6320447681\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.95 - 2.73i)T \)
5 \( 1 \)
good3 \( 1 + 18.7T + 243T^{2} \)
7 \( 1 - 207. iT - 1.68e4T^{2} \)
11 \( 1 + 248. iT - 1.61e5T^{2} \)
13 \( 1 - 532.T + 3.71e5T^{2} \)
17 \( 1 + 710. iT - 1.41e6T^{2} \)
19 \( 1 - 1.34e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.54e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.82e3T + 2.86e7T^{2} \)
37 \( 1 - 556.T + 6.93e7T^{2} \)
41 \( 1 - 1.66e4T + 1.15e8T^{2} \)
43 \( 1 + 2.08e4T + 1.47e8T^{2} \)
47 \( 1 - 1.87e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.67e4T + 4.18e8T^{2} \)
59 \( 1 - 3.03e4iT - 7.14e8T^{2} \)
61 \( 1 + 7.33e3iT - 8.44e8T^{2} \)
67 \( 1 + 3.46e4T + 1.35e9T^{2} \)
71 \( 1 - 5.00e4T + 1.80e9T^{2} \)
73 \( 1 - 6.32e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.29e4T + 3.07e9T^{2} \)
83 \( 1 - 1.01e5T + 3.93e9T^{2} \)
89 \( 1 + 8.92e4T + 5.58e9T^{2} \)
97 \( 1 + 7.24e4iT - 8.58e9T^{2} \)
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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

−11.64295826803420615930613190555, −11.00209134228945695716523191661, −9.888745155060000592020367176551, −8.794569109393592631612316624180, −8.060750114013183327548595908359, −6.22575334325699089931604896881, −6.10077841989519104098905522140, −4.96987529283769573434626367793, −2.58930687560135412288390015339, −0.911998341103686801916913549049, 0.43369149105034070539826422871, 1.41316631981710313506222585333, 3.49432927815520254758478629571, 4.70994357066300722483523903590, 6.38780117457582098959791489739, 7.11860605966124204087737202382, 8.228271735054521656635743736333, 9.608350003585458628849690372655, 10.57338721282384229857882542087, 11.00333318009930949659451000423

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