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Properties

Label 2-20e2-100.23-c1-0-14
Degree $2$
Conductor $400$
Sign $-0.425 - 0.904i$
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  + (−1.11 − 2.19i)3-s − 2.23·5-s + (−2.71 − 2.71i)7-s + (−1.79 + 2.47i)9-s + (1.87 + 2.57i)11-s + (0.951 + 6.00i)13-s + (2.49 + 4.90i)15-s + (−2.48 − 1.26i)17-s + (0.812 + 2.49i)19-s + (−2.91 + 8.97i)21-s + (0.902 − 5.70i)23-s + 5.00·25-s + (0.147 + 0.0232i)27-s + (−3.50 − 1.13i)29-s + (−6.62 + 2.15i)31-s + ⋯
L(s)  = 1  + (−0.645 − 1.26i)3-s − 0.999·5-s + (−1.02 − 1.02i)7-s + (−0.599 + 0.825i)9-s + (0.564 + 0.777i)11-s + (0.263 + 1.66i)13-s + (0.645 + 1.26i)15-s + (−0.603 − 0.307i)17-s + (0.186 + 0.573i)19-s + (−0.636 + 1.95i)21-s + (0.188 − 1.18i)23-s + 1.00·25-s + (0.0283 + 0.00448i)27-s + (−0.650 − 0.211i)29-s + (−1.19 + 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.425 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00972863 + 0.0153298i\)
\(L(\frac12)\) \(\approx\) \(0.00972863 + 0.0153298i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T \)
good3 \( 1 + (1.11 + 2.19i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (2.71 + 2.71i)T + 7iT^{2} \)
11 \( 1 + (-1.87 - 2.57i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.951 - 6.00i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.48 + 1.26i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.812 - 2.49i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.902 + 5.70i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (3.50 + 1.13i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.62 - 2.15i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.82 - 1.39i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-1.43 - 1.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (3.91 - 3.91i)T - 43iT^{2} \)
47 \( 1 + (7.95 - 4.05i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.97 + 3.04i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (2.77 + 2.01i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.82 + 2.05i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.49 + 4.90i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (14.7 + 4.78i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.39 - 1.01i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.37 - 4.22i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.26 + 4.20i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (6.66 + 9.16i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.29 + 6.46i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.99086050026247399373181809570, −9.775747057703549450089973093952, −8.712143536429644961829672556287, −7.37493069250104211287951978434, −6.91260473074431602236461472662, −6.39378082789076209444276785398, −4.57666293674348714170145968905, −3.63869644047795935386928329813, −1.65252445503021153057218690092, −0.01284117382565404769580890428, 3.21837342195971962974234095517, 3.78120736488925472469064980457, 5.28710774154954589781051267280, 5.81740423182166237438363290733, 7.15806734584278001385695173679, 8.549761280981335585804371041239, 9.167247422070336572087126812763, 10.18287405239466294327686100581, 11.04683567567441127391094213940, 11.60078453910274893510769880258

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