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Properties

Label 2-20e2-80.3-c1-0-17
Degree $2$
Conductor $400$
Sign $0.997 - 0.0720i$
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  + (−1.19 + 0.759i)2-s + 1.39·3-s + (0.846 − 1.81i)4-s + (−1.66 + 1.05i)6-s + (2.13 − 2.13i)7-s + (0.366 + 2.80i)8-s − 1.05·9-s + (2.17 + 2.17i)11-s + (1.17 − 2.52i)12-s − 1.54i·13-s + (−0.925 + 4.16i)14-s + (−2.56 − 3.06i)16-s + (3.86 − 3.86i)17-s + (1.26 − 0.804i)18-s + (0.0136 + 0.0136i)19-s + ⋯
L(s)  = 1  + (−0.843 + 0.536i)2-s + 0.804·3-s + (0.423 − 0.905i)4-s + (−0.678 + 0.431i)6-s + (0.806 − 0.806i)7-s + (0.129 + 0.991i)8-s − 0.353·9-s + (0.654 + 0.654i)11-s + (0.340 − 0.728i)12-s − 0.428i·13-s + (−0.247 + 1.11i)14-s + (−0.641 − 0.766i)16-s + (0.937 − 0.937i)17-s + (0.297 − 0.189i)18-s + (0.00313 + 0.00313i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.997 - 0.0720i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.997 - 0.0720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30503 + 0.0470951i\)
\(L(\frac12)\) \(\approx\) \(1.30503 + 0.0470951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.759i)T \)
5 \( 1 \)
good3 \( 1 - 1.39T + 3T^{2} \)
7 \( 1 + (-2.13 + 2.13i)T - 7iT^{2} \)
11 \( 1 + (-2.17 - 2.17i)T + 11iT^{2} \)
13 \( 1 + 1.54iT - 13T^{2} \)
17 \( 1 + (-3.86 + 3.86i)T - 17iT^{2} \)
19 \( 1 + (-0.0136 - 0.0136i)T + 19iT^{2} \)
23 \( 1 + (-3.15 - 3.15i)T + 23iT^{2} \)
29 \( 1 + (-3.33 + 3.33i)T - 29iT^{2} \)
31 \( 1 + 8.92iT - 31T^{2} \)
37 \( 1 - 7.24iT - 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 2.02iT - 43T^{2} \)
47 \( 1 + (3.34 + 3.34i)T + 47iT^{2} \)
53 \( 1 - 7.30T + 53T^{2} \)
59 \( 1 + (3.52 - 3.52i)T - 59iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + 61iT^{2} \)
67 \( 1 - 0.748iT - 67T^{2} \)
71 \( 1 + 0.269T + 71T^{2} \)
73 \( 1 + (0.811 - 0.811i)T - 73iT^{2} \)
79 \( 1 + 2.80T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (6.33 - 6.33i)T - 97iT^{2} \)
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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

−11.23705914141118372077366079423, −9.952763486345896652555616352875, −9.505002673527219263488329930720, −8.319576350158978820786068672314, −7.78009114295909982932262254173, −6.96798757203015423794531673659, −5.64136615096924770636219133669, −4.45499483011858328892079541804, −2.82572316775716980999084833501, −1.26315288782787027705039283156, 1.58234729520975176758554631532, 2.80886378556081430185793603761, 3.84177828019510296564349514309, 5.50949145856586194392272645557, 6.83708798648603089024748485896, 8.036749300191444228948951837328, 8.743130970009082456342545066629, 9.027706173234136941218813155795, 10.37747639637510164248412533921, 11.17430955362481170682957775618

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