Properties

Label 2-20e2-400.163-c1-0-45
Degree $2$
Conductor $400$
Sign $0.671 + 0.741i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.00394i)2-s + (−0.605 − 1.86i)3-s + (1.99 − 0.0111i)4-s + (2.22 + 0.190i)5-s + (−0.864 − 2.63i)6-s + (−0.938 + 0.938i)7-s + (2.82 − 0.0236i)8-s + (−0.683 + 0.496i)9-s + (3.15 + 0.261i)10-s + (0.158 + 1.00i)11-s + (−1.23 − 3.72i)12-s + (−0.530 − 0.730i)13-s + (−1.32 + 1.33i)14-s + (−0.994 − 4.27i)15-s + (3.99 − 0.0446i)16-s + (−0.416 − 0.816i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.00279i)2-s + (−0.349 − 1.07i)3-s + (0.999 − 0.00558i)4-s + (0.996 + 0.0853i)5-s + (−0.352 − 1.07i)6-s + (−0.354 + 0.354i)7-s + (0.999 − 0.00837i)8-s + (−0.227 + 0.165i)9-s + (0.996 + 0.0825i)10-s + (0.0477 + 0.301i)11-s + (−0.355 − 1.07i)12-s + (−0.147 − 0.202i)13-s + (−0.353 + 0.355i)14-s + (−0.256 − 1.10i)15-s + (0.999 − 0.0111i)16-s + (−0.100 − 0.198i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.671 + 0.741i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.671 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32325 - 1.03079i\)
\(L(\frac12)\) \(\approx\) \(2.32325 - 1.03079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.00394i)T \)
5 \( 1 + (-2.22 - 0.190i)T \)
good3 \( 1 + (0.605 + 1.86i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.938 - 0.938i)T - 7iT^{2} \)
11 \( 1 + (-0.158 - 1.00i)T + (-10.4 + 3.39i)T^{2} \)
13 \( 1 + (0.530 + 0.730i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.416 + 0.816i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (6.10 - 3.11i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (0.319 + 2.01i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (5.39 + 2.74i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (2.13 + 0.693i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.80 - 3.85i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.36 - 7.38i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.776iT - 43T^{2} \)
47 \( 1 + (-0.365 + 0.716i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (2.92 + 9.00i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.61 - 10.1i)T + (-56.1 - 18.2i)T^{2} \)
61 \( 1 + (-0.0420 - 0.265i)T + (-58.0 + 18.8i)T^{2} \)
67 \( 1 + (-3.29 - 1.06i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.31 - 10.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (12.5 - 1.98i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (0.916 + 2.82i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.87 + 11.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.46 - 3.24i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.05 + 1.55i)T + (57.0 + 78.4i)T^{2} \)
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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

−11.45360484857874541414990882598, −10.43531103528123173625158857123, −9.513192732331944829945989223175, −8.052460027445920121613883947960, −6.98791058007874462248712924351, −6.24191301472754757967279439995, −5.69221752726201895696212257718, −4.31047529399387257764447446341, −2.64942763650724473282206607738, −1.67984867860066921541218931162, 2.10777091978774759753688094602, 3.61187616149674642968600802222, 4.54860560694868810643799325765, 5.46623775056939355789484887206, 6.29116442888561408847383437202, 7.31803850520206524046648862177, 8.956773507881835754992289695466, 9.807550858469911703597642467430, 10.78500873047080838416461300558, 11.02668350744247649701431531529

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