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Properties

Label 2-20e2-80.43-c1-0-30
Degree $2$
Conductor $400$
Sign $0.0862 + 0.996i$
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  + (1.31 − 0.516i)2-s − 1.28i·3-s + (1.46 − 1.36i)4-s + (−0.662 − 1.68i)6-s + (1.13 − 1.13i)7-s + (1.22 − 2.54i)8-s + 1.35·9-s + (−2.32 + 2.32i)11-s + (−1.74 − 1.87i)12-s − 1.36·13-s + (0.911 − 2.08i)14-s + (0.297 − 3.98i)16-s + (−5.25 + 5.25i)17-s + (1.78 − 0.702i)18-s + (3.69 − 3.69i)19-s + ⋯
L(s)  = 1  + (0.930 − 0.365i)2-s − 0.739i·3-s + (0.732 − 0.680i)4-s + (−0.270 − 0.688i)6-s + (0.430 − 0.430i)7-s + (0.433 − 0.901i)8-s + 0.452·9-s + (−0.700 + 0.700i)11-s + (−0.503 − 0.542i)12-s − 0.378·13-s + (0.243 − 0.558i)14-s + (0.0744 − 0.997i)16-s + (−1.27 + 1.27i)17-s + (0.421 − 0.165i)18-s + (0.848 − 0.848i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0862 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0862 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.0862 + 0.996i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.0862 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78534 - 1.63739i\)
\(L(\frac12)\) \(\approx\) \(1.78534 - 1.63739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.516i)T \)
5 \( 1 \)
good3 \( 1 + 1.28iT - 3T^{2} \)
7 \( 1 + (-1.13 + 1.13i)T - 7iT^{2} \)
11 \( 1 + (2.32 - 2.32i)T - 11iT^{2} \)
13 \( 1 + 1.36T + 13T^{2} \)
17 \( 1 + (5.25 - 5.25i)T - 17iT^{2} \)
19 \( 1 + (-3.69 + 3.69i)T - 19iT^{2} \)
23 \( 1 + (-0.911 - 0.911i)T + 23iT^{2} \)
29 \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \)
31 \( 1 - 0.242iT - 31T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 + 9.04T + 43T^{2} \)
47 \( 1 + (-7.87 - 7.87i)T + 47iT^{2} \)
53 \( 1 - 5.80iT - 53T^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 3.26iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (-1.63 + 1.63i)T - 97iT^{2} \)
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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

−11.13530532448940231751475844178, −10.48753315362565571793953510947, −9.462072289670013861255344808519, −7.911903548879893184531550329464, −7.14916193869394295044425084078, −6.35882258564903676368964970654, −4.98238737548898322261738800262, −4.23509339891227140283528026165, −2.61696234844557001923674639050, −1.45260300411969646177997292723, 2.39828955973848267959237236325, 3.63053772885222574042458945924, 4.82412534222774360985926082260, 5.34642673886830725146195326613, 6.65781985840490501993183833052, 7.65642505555676232281252886253, 8.630003072133210017230012219086, 9.751478317607647368458180911844, 10.78438758522867151756499152446, 11.54335699469557527242818331718

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