Properties

Label 2-20e2-25.21-c1-0-11
Degree $2$
Conductor $400$
Sign $0.535 + 0.844i$
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  + (0.809 + 0.587i)3-s + (−0.690 − 2.12i)5-s − 0.618·7-s + (−0.618 − 1.90i)9-s + (1.61 − 4.97i)11-s + (0.572 + 1.76i)13-s + (0.690 − 2.12i)15-s + (4.23 − 3.07i)17-s + (0.690 − 0.502i)19-s + (−0.500 − 0.363i)21-s + (−1.16 + 3.57i)23-s + (−4.04 + 2.93i)25-s + (1.54 − 4.75i)27-s + (2.92 + 2.12i)29-s + (−2.42 + 1.76i)31-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (−0.309 − 0.951i)5-s − 0.233·7-s + (−0.206 − 0.634i)9-s + (0.487 − 1.50i)11-s + (0.158 + 0.489i)13-s + (0.178 − 0.549i)15-s + (1.02 − 0.746i)17-s + (0.158 − 0.115i)19-s + (−0.109 − 0.0792i)21-s + (−0.242 + 0.746i)23-s + (−0.809 + 0.587i)25-s + (0.297 − 0.915i)27-s + (0.543 + 0.394i)29-s + (−0.435 + 0.316i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.535 + 0.844i$
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: \((2,\ 400,\ (\ :1/2),\ 0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26113 - 0.693313i\)
\(L(\frac12)\) \(\approx\) \(1.26113 - 0.693313i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 + 2.12i)T \)
good3 \( 1 + (-0.809 - 0.587i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 0.618T + 7T^{2} \)
11 \( 1 + (-1.61 + 4.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.572 - 1.76i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.23 + 3.07i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.690 + 0.502i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.16 - 3.57i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.92 - 2.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.42 - 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0729 + 0.224i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.236 + 0.726i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.363i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.80 - 2.04i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.35 - 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.69 + 8.28i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.85 + 2.80i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.35 + 3.88i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.78 - 8.55i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.54 + 4.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.04 - 3.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.76 - 8.50i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.11 - 2.26i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.37928188866988108406516717824, −9.975418250189045427373611926248, −9.090227335867207349068232064586, −8.675984357603885905740269507592, −7.59449742170499026996537561661, −6.24782377779695983774672907165, −5.29736920317658882506503356110, −3.93613951698086123769483032382, −3.17170174766898298690705606322, −0.971155834084136433971578650085, 1.98819394678751910442890975151, 3.14731942388278701332579685067, 4.35483719039384631653525408076, 5.81593469294710744862185196303, 6.93661553329277548249277608163, 7.64964169634672013110252444161, 8.444063948624921235690630365980, 9.843748913864362287029030510654, 10.35490414906394033419346201061, 11.41535371494637226609520112039

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