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Properties

Label 2-20e2-100.27-c1-0-14
Degree $2$
Conductor $400$
Sign $0.498 + 0.866i$
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  + (1.94 − 0.990i)3-s + (1.03 − 1.98i)5-s + (1.48 − 1.48i)7-s + (1.03 − 1.42i)9-s + (−0.172 − 0.236i)11-s + (−4.46 + 0.706i)13-s + (0.0547 − 4.88i)15-s + (−1.99 + 3.92i)17-s + (0.873 + 2.68i)19-s + (1.41 − 4.35i)21-s + (3.88 + 0.615i)23-s + (−2.84 − 4.11i)25-s + (−0.421 + 2.66i)27-s + (−5.50 − 1.78i)29-s + (9.13 − 2.96i)31-s + ⋯
L(s)  = 1  + (1.12 − 0.572i)3-s + (0.463 − 0.885i)5-s + (0.560 − 0.560i)7-s + (0.345 − 0.475i)9-s + (−0.0518 − 0.0714i)11-s + (−1.23 + 0.196i)13-s + (0.0141 − 1.26i)15-s + (−0.484 + 0.950i)17-s + (0.200 + 0.616i)19-s + (0.308 − 0.950i)21-s + (0.810 + 0.128i)23-s + (−0.569 − 0.822i)25-s + (−0.0811 + 0.512i)27-s + (−1.02 − 0.331i)29-s + (1.64 − 0.533i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.498 + 0.866i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78631 - 1.03312i\)
\(L(\frac12)\) \(\approx\) \(1.78631 - 1.03312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 + 1.98i)T \)
good3 \( 1 + (-1.94 + 0.990i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-1.48 + 1.48i)T - 7iT^{2} \)
11 \( 1 + (0.172 + 0.236i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.46 - 0.706i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.99 - 3.92i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-0.873 - 2.68i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.88 - 0.615i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (5.50 + 1.78i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.13 + 2.96i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.750 - 4.73i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-5.38 - 3.91i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.39 - 5.39i)T + 43iT^{2} \)
47 \( 1 + (3.24 + 6.36i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (3.61 + 7.10i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (2.43 + 1.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.58 + 6.96i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (8.32 + 4.24i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (0.108 + 0.0353i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.879 - 5.55i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (4.32 - 13.3i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.48 + 14.6i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-6.38 - 8.78i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.01 - 1.02i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.15559549443059610619375844530, −9.921201572946360496110250805358, −9.238334640022060600344108242381, −8.136653232046955692583912235982, −7.81182196089365055904955238005, −6.53629089943253889355035058539, −5.16001873475884813354336474785, −4.14500829349107932872878517577, −2.55797224378132654123185607044, −1.45474549360290967106740054848, 2.41156688504720061859795418932, 2.92488824550610112115584873825, 4.45969529115110917216741059412, 5.51687265303169344076690698053, 6.95755356152821835209577006623, 7.72680331092961605302770580838, 8.984530295924776770520380091692, 9.407815006979187143001539027649, 10.37595843252524476468312381895, 11.27601493282348539310886309512

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