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Properties

Label 2-20e2-25.21-c1-0-4
Degree $2$
Conductor $400$
Sign $0.968 - 0.248i$
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  + (−0.309 − 0.224i)3-s + (1.80 + 1.31i)5-s + 3·7-s + (−0.881 − 2.71i)9-s + (−1.30 + 4.02i)11-s + (0.309 + 0.951i)13-s + (−0.263 − 0.812i)15-s + (−0.927 + 0.673i)17-s + (4.73 − 3.44i)19-s + (−0.927 − 0.673i)21-s + (0.545 − 1.67i)23-s + (1.54 + 4.75i)25-s + (−0.690 + 2.12i)27-s + (7.66 + 5.56i)29-s + (−0.190 + 0.138i)31-s + ⋯
L(s)  = 1  + (−0.178 − 0.129i)3-s + (0.809 + 0.587i)5-s + 1.13·7-s + (−0.293 − 0.904i)9-s + (−0.394 + 1.21i)11-s + (0.0857 + 0.263i)13-s + (−0.0681 − 0.209i)15-s + (−0.224 + 0.163i)17-s + (1.08 − 0.789i)19-s + (−0.202 − 0.146i)21-s + (0.113 − 0.349i)23-s + (0.309 + 0.951i)25-s + (−0.132 + 0.409i)27-s + (1.42 + 1.03i)29-s + (−0.0343 + 0.0249i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58779 + 0.200584i\)
\(L(\frac12)\) \(\approx\) \(1.58779 + 0.200584i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.224i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + (1.30 - 4.02i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.927 - 0.673i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.73 + 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.545 + 1.67i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-7.66 - 5.56i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.190 - 0.138i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.57 + 7.91i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.454 - 1.40i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + (9.66 + 7.02i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (8.47 + 6.15i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.38 - 4.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.28 - 6.01i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.42 + 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.38 + 7.33i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.85 + 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.66 - 2.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.38 + 4.25i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (7.73 + 5.62i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.28767777793164712100382808485, −10.45212053739729166873565152364, −9.530283083374762117789673921524, −8.675904754515536276199475102082, −7.35850653024209585994170238437, −6.70161753087092963578358559930, −5.49686846298366543342347085464, −4.59539011608127370330086397284, −2.93036188926883865560803326429, −1.61160961171874541688535743557, 1.37281552333712775917710999015, 2.85297794061122481637129880291, 4.67342551298250568267422256156, 5.35278248576466047542935991219, 6.18207247947277812704345338023, 7.997254195665320768416219576909, 8.197651579182130489122161693468, 9.455667784301405522413015538065, 10.43496249580977962789920097219, 11.18209879571758484492599673293

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