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Properties

Label 2-20e2-25.19-c1-0-5
Degree $2$
Conductor $400$
Sign $0.320 - 0.947i$
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 + 1.81i)3-s + (2.19 + 0.432i)5-s + 1.85i·7-s + (−0.624 + 1.92i)9-s + (0.681 + 2.09i)11-s + (−4.55 − 1.47i)13-s + (2.10 + 4.54i)15-s + (3.76 − 5.17i)17-s + (−3.01 − 2.18i)19-s + (−3.37 + 2.44i)21-s + (0.905 − 0.294i)23-s + (4.62 + 1.89i)25-s + (2.08 − 0.677i)27-s + (−4.71 + 3.42i)29-s + (−5.42 − 3.94i)31-s + ⋯
L(s)  = 1  + (0.760 + 1.04i)3-s + (0.981 + 0.193i)5-s + 0.702i·7-s + (−0.208 + 0.640i)9-s + (0.205 + 0.632i)11-s + (−1.26 − 0.410i)13-s + (0.543 + 1.17i)15-s + (0.912 − 1.25i)17-s + (−0.690 − 0.501i)19-s + (−0.735 + 0.534i)21-s + (0.188 − 0.0613i)23-s + (0.925 + 0.379i)25-s + (0.401 − 0.130i)27-s + (−0.875 + 0.635i)29-s + (−0.974 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58770 + 1.13864i\)
\(L(\frac12)\) \(\approx\) \(1.58770 + 1.13864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 0.432i)T \)
good3 \( 1 + (-1.31 - 1.81i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 1.85iT - 7T^{2} \)
11 \( 1 + (-0.681 - 2.09i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (4.55 + 1.47i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.76 + 5.17i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.01 + 2.18i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.905 + 0.294i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (4.71 - 3.42i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.42 + 3.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.69 - 3.14i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.82 - 5.63i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.96iT - 43T^{2} \)
47 \( 1 + (5.93 + 8.16i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.46 + 3.39i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.86 - 5.73i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.13 + 3.49i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.45 + 11.6i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.82 + 5.68i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.43 + 1.11i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.13 - 6.63i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.08 + 5.62i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.23 + 9.96i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.275 - 0.379i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.30168013240661315648785233189, −10.15078168810430520494440893864, −9.513329720530762687329862304217, −9.218811735828105961461063354142, −7.88448105584927543759585836561, −6.75234320474159363613566342902, −5.40022712144954876108510453652, −4.69287424095010337300717996346, −3.14067806977795165504308495475, −2.28514943398822762801082156367, 1.41197748728801755065443900511, 2.45193901396339050855241719125, 3.92728542943427841604379899167, 5.48158322332265141850482173242, 6.49476245249323919896548169197, 7.42957172380289241122512258791, 8.208806364608772261606919833156, 9.204636222119928996880952907929, 10.08515944721085353085846536465, 10.99465325101487791731271604438

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