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Properties

Label 2-20e2-16.11-c2-0-56
Degree $2$
Conductor $400$
Sign $0.948 + 0.317i$
Analytic cond. $10.8992$
Root an. cond. $3.30139$
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  + (1.99 − 0.0345i)2-s + (0.374 + 0.374i)3-s + (3.99 − 0.138i)4-s + (0.762 + 0.736i)6-s − 2.42·7-s + (7.98 − 0.414i)8-s − 8.71i·9-s + (13.8 − 13.8i)11-s + (1.55 + 1.44i)12-s + (−8.90 + 8.90i)13-s + (−4.84 + 0.0838i)14-s + (15.9 − 1.10i)16-s + 27.7·17-s + (−0.301 − 17.4i)18-s + (7.49 + 7.49i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0172i)2-s + (0.124 + 0.124i)3-s + (0.999 − 0.0345i)4-s + (0.127 + 0.122i)6-s − 0.346·7-s + (0.998 − 0.0518i)8-s − 0.968i·9-s + (1.25 − 1.25i)11-s + (0.129 + 0.120i)12-s + (−0.684 + 0.684i)13-s + (−0.346 + 0.00598i)14-s + (0.997 − 0.0690i)16-s + 1.63·17-s + (−0.0167 − 0.968i)18-s + (0.394 + 0.394i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.948 + 0.317i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ 0.948 + 0.317i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.45292 - 0.563544i\)
\(L(\frac12)\) \(\approx\) \(3.45292 - 0.563544i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.0345i)T \)
5 \( 1 \)
good3 \( 1 + (-0.374 - 0.374i)T + 9iT^{2} \)
7 \( 1 + 2.42T + 49T^{2} \)
11 \( 1 + (-13.8 + 13.8i)T - 121iT^{2} \)
13 \( 1 + (8.90 - 8.90i)T - 169iT^{2} \)
17 \( 1 - 27.7T + 289T^{2} \)
19 \( 1 + (-7.49 - 7.49i)T + 361iT^{2} \)
23 \( 1 + 13.5T + 529T^{2} \)
29 \( 1 + (10.5 - 10.5i)T - 841iT^{2} \)
31 \( 1 - 46.4iT - 961T^{2} \)
37 \( 1 + (-4.68 - 4.68i)T + 1.36e3iT^{2} \)
41 \( 1 + 38.6iT - 1.68e3T^{2} \)
43 \( 1 + (45.6 - 45.6i)T - 1.84e3iT^{2} \)
47 \( 1 + 35.3iT - 2.20e3T^{2} \)
53 \( 1 + (61.0 + 61.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (9.21 - 9.21i)T - 3.48e3iT^{2} \)
61 \( 1 + (-20.3 + 20.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (3.47 + 3.47i)T + 4.48e3iT^{2} \)
71 \( 1 + 8.72T + 5.04e3T^{2} \)
73 \( 1 + 23.1iT - 5.32e3T^{2} \)
79 \( 1 - 73.9iT - 6.24e3T^{2} \)
83 \( 1 + (-50.7 - 50.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 51.4iT - 7.92e3T^{2} \)
97 \( 1 + 82.8T + 9.40e3T^{2} \)
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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

−11.39421083198600035853609725797, −10.11258280640149714956948095056, −9.347744639223932401620757434679, −8.153213439709344014271466523920, −6.86541527444929331483182347890, −6.23111015263353296481885178923, −5.18993965564240854611426292584, −3.71183181325594656417562524510, −3.28437387398542817530672807467, −1.33248410140073851704679532409, 1.70547312167473437047926116642, 2.97325390812120304917315401250, 4.22464126953937803835365727216, 5.18988478080210698935045704680, 6.21807683200579580590929302909, 7.39223124791257412845391058015, 7.84679971819234073910762655032, 9.622480312987490661184026039086, 10.16084215530528498318453758988, 11.41430552608508094183624113699

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