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Properties

Label 2-20e2-80.19-c2-0-25
Degree $2$
Conductor $400$
Sign $-0.997 + 0.0716i$
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  + (0.390 + 1.96i)2-s + (3.81 + 3.81i)3-s + (−3.69 + 1.53i)4-s + (−5.99 + 8.97i)6-s + 7.06i·7-s + (−4.44 − 6.64i)8-s + 20.1i·9-s + (10.3 + 10.3i)11-s + (−19.9 − 8.25i)12-s + (5.72 − 5.72i)13-s + (−13.8 + 2.75i)14-s + (11.3 − 11.3i)16-s − 20.9i·17-s + (−39.4 + 7.85i)18-s + (7.58 − 7.58i)19-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)2-s + (1.27 + 1.27i)3-s + (−0.923 + 0.383i)4-s + (−0.999 + 1.49i)6-s + 1.00i·7-s + (−0.556 − 0.831i)8-s + 2.23i·9-s + (0.942 + 0.942i)11-s + (−1.66 − 0.687i)12-s + (0.440 − 0.440i)13-s + (−0.989 + 0.197i)14-s + (0.706 − 0.707i)16-s − 1.23i·17-s + (−2.19 + 0.436i)18-s + (0.398 − 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.997 + 0.0716i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ -0.997 + 0.0716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0939103 - 2.61954i\)
\(L(\frac12)\) \(\approx\) \(0.0939103 - 2.61954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.390 - 1.96i)T \)
5 \( 1 \)
good3 \( 1 + (-3.81 - 3.81i)T + 9iT^{2} \)
7 \( 1 - 7.06iT - 49T^{2} \)
11 \( 1 + (-10.3 - 10.3i)T + 121iT^{2} \)
13 \( 1 + (-5.72 + 5.72i)T - 169iT^{2} \)
17 \( 1 + 20.9iT - 289T^{2} \)
19 \( 1 + (-7.58 + 7.58i)T - 361iT^{2} \)
23 \( 1 + 13.7iT - 529T^{2} \)
29 \( 1 + (32.5 + 32.5i)T + 841iT^{2} \)
31 \( 1 - 26.0iT - 961T^{2} \)
37 \( 1 + (19.6 + 19.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 + (1.30 - 1.30i)T - 1.84e3iT^{2} \)
47 \( 1 - 25.9T + 2.20e3T^{2} \)
53 \( 1 + (8.62 + 8.62i)T + 2.80e3iT^{2} \)
59 \( 1 + (-43.8 - 43.8i)T + 3.48e3iT^{2} \)
61 \( 1 + (26.3 + 26.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (-62.9 - 62.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 67.9T + 5.04e3T^{2} \)
73 \( 1 - 14.0T + 5.32e3T^{2} \)
79 \( 1 + 11.0iT - 6.24e3T^{2} \)
83 \( 1 + (70.2 + 70.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 161. iT - 7.92e3T^{2} \)
97 \( 1 + 125. iT - 9.40e3T^{2} \)
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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

−11.54108586828092242449560526342, −10.08321938803558290560712727286, −9.317267989231969392611315448733, −8.943745945276633151540403637314, −8.007780871485078603411877874372, −6.97968038405988281586945103210, −5.53931401254901668684320371753, −4.66367554753454976516581635926, −3.71597178153220296972658075182, −2.57579570203253857881638453607, 1.03677406642169974223651207530, 1.85627030737074293394718287059, 3.49871155501774057970703809635, 3.82185376222455711030709469910, 5.90555410597601390323367997330, 6.95953991827746568040015310824, 8.030581134530615984642700749727, 8.774477564216925137949046424933, 9.527466998888861113994799038684, 10.75159895095410148045241930949

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