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Properties

Label 2-200-40.13-c2-0-25
Degree $2$
Conductor $200$
Sign $0.516 + 0.856i$
Analytic cond. $5.44960$
Root an. cond. $2.33443$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 + 0.993i)2-s + (3.30 − 3.30i)3-s + (2.02 − 3.44i)4-s + (−2.44 + 9.01i)6-s + (6.68 − 6.68i)7-s + (−0.0871 + 7.99i)8-s − 12.7i·9-s + 13.9i·11-s + (−4.70 − 18.0i)12-s + (9.57 − 9.57i)13-s + (−4.96 + 18.2i)14-s + (−7.79 − 13.9i)16-s + (2.45 − 2.45i)17-s + (12.7 + 22.2i)18-s − 30.1·19-s + ⋯
L(s)  = 1  + (−0.867 + 0.496i)2-s + (1.10 − 1.10i)3-s + (0.506 − 0.862i)4-s + (−0.408 + 1.50i)6-s + (0.955 − 0.955i)7-s + (−0.0108 + 0.999i)8-s − 1.42i·9-s + 1.27i·11-s + (−0.391 − 1.50i)12-s + (0.736 − 0.736i)13-s + (−0.354 + 1.30i)14-s + (−0.487 − 0.873i)16-s + (0.144 − 0.144i)17-s + (0.706 + 1.23i)18-s − 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(5.44960\)
Root analytic conductor: \(2.33443\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :1),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.36736 - 0.772143i\)
\(L(\frac12)\) \(\approx\) \(1.36736 - 0.772143i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 - 0.993i)T \)
5 \( 1 \)
good3 \( 1 + (-3.30 + 3.30i)T - 9iT^{2} \)
7 \( 1 + (-6.68 + 6.68i)T - 49iT^{2} \)
11 \( 1 - 13.9iT - 121T^{2} \)
13 \( 1 + (-9.57 + 9.57i)T - 169iT^{2} \)
17 \( 1 + (-2.45 + 2.45i)T - 289iT^{2} \)
19 \( 1 + 30.1T + 361T^{2} \)
23 \( 1 + (15.1 + 15.1i)T + 529iT^{2} \)
29 \( 1 - 11.7T + 841T^{2} \)
31 \( 1 - 47.1T + 961T^{2} \)
37 \( 1 + (7.26 + 7.26i)T + 1.36e3iT^{2} \)
41 \( 1 - 15.7T + 1.68e3T^{2} \)
43 \( 1 + (34.3 - 34.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-23.6 + 23.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (31.3 - 31.3i)T - 2.80e3iT^{2} \)
59 \( 1 + 2.22T + 3.48e3T^{2} \)
61 \( 1 - 55.8iT - 3.72e3T^{2} \)
67 \( 1 + (-17.1 - 17.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 78.3T + 5.04e3T^{2} \)
73 \( 1 + (-31.4 - 31.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 95.1iT - 6.24e3T^{2} \)
83 \( 1 + (1.30 - 1.30i)T - 6.88e3iT^{2} \)
89 \( 1 - 45.1iT - 7.92e3T^{2} \)
97 \( 1 + (-60.8 + 60.8i)T - 9.40e3iT^{2} \)
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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

−12.16911050306408626884498487579, −10.79989453785002953241170728838, −10.00138869038945743190602667201, −8.562509740052374658052879523627, −8.083815473823418851709993464012, −7.27451109942765353597461957429, −6.37832300465951529768580509683, −4.48587068143399031266444975298, −2.35149397895326676984197130462, −1.19210831033667518497523131466, 2.01316438053977480269899315750, 3.28360521195966138309002796787, 4.41834726745131868183338629374, 6.20261479652180608762932541627, 8.223380125227595598395252727538, 8.457377750487549145881301798915, 9.223756921184261785595435598110, 10.34832440906580609795201447354, 11.16309030477243574396190734378, 11.99882273879659588635174550736

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