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Properties

Label 2-20e2-25.16-c1-0-12
Degree $2$
Conductor $400$
Sign $-0.728 - 0.684i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 − 2.12i)3-s + (−1.80 + 1.31i)5-s − 0.381·7-s + (−1.61 + 1.17i)9-s + (−2.61 − 1.90i)11-s + (−4.54 + 3.30i)13-s + (4.04 + 2.93i)15-s + (0.236 − 0.726i)17-s + (−1.66 + 5.11i)19-s + (0.263 + 0.812i)21-s + (−2.80 − 2.04i)23-s + (1.54 − 4.75i)25-s + (−1.80 − 1.31i)27-s + (2.04 + 6.29i)29-s + (−2.69 + 8.28i)31-s + ⋯
L(s)  = 1  + (−0.398 − 1.22i)3-s + (−0.809 + 0.587i)5-s − 0.144·7-s + (−0.539 + 0.391i)9-s + (−0.789 − 0.573i)11-s + (−1.26 + 0.915i)13-s + (1.04 + 0.758i)15-s + (0.0572 − 0.176i)17-s + (−0.381 + 1.17i)19-s + (0.0575 + 0.177i)21-s + (−0.585 − 0.425i)23-s + (0.309 − 0.951i)25-s + (−0.348 − 0.252i)27-s + (0.379 + 1.16i)29-s + (−0.483 + 1.48i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.728 - 0.684i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.31i)T \)
good3 \( 1 + (0.690 + 2.12i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + 0.381T + 7T^{2} \)
11 \( 1 + (2.61 + 1.90i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.54 - 3.30i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.236 + 0.726i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.66 - 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.80 + 2.04i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.04 - 6.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.69 - 8.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.04 + 2.21i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.23 + 4.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.61T + 43T^{2} \)
47 \( 1 + (2.11 + 6.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.07 + 9.45i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.73 + 2.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.42 + 5.39i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.0901 - 0.277i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.97 - 6.06i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.89 + 6.46i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.19 + 9.82i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.16 + 12.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.23 + 2.35i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.64 - 17.3i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.86891334858678925345945765770, −10.01135078805604890700907913941, −8.555081206714864171852940649764, −7.67737509316605092104011835776, −7.00525397376144355411231935623, −6.22460029739469134102885203736, −4.89791447146227460277119082518, −3.40751764460892720027176593480, −2.02185492668020606538798064378, 0, 2.78602905350476929909012696206, 4.28912197314745230527784644345, 4.77311958379456484569000040560, 5.78298659771370944944734199231, 7.44017891647816312287291310827, 8.079151820143797220713021602007, 9.476950146743473327888390796445, 9.901257840969248065535578135897, 10.91459238569389821652511251387

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