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Properties

Label 2-20e2-80.29-c1-0-18
Degree $2$
Conductor $400$
Sign $0.575 - 0.818i$
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 + 0.507i)2-s + (−0.0623 + 0.0623i)3-s + (1.48 + 1.34i)4-s + (−0.113 + 0.0506i)6-s + 0.375·7-s + (1.27 + 2.52i)8-s + 2.99i·9-s + (2.36 − 2.36i)11-s + (−0.176 + 0.00895i)12-s + (−1.76 + 1.76i)13-s + (0.496 + 0.190i)14-s + (0.405 + 3.97i)16-s − 4.64i·17-s + (−1.51 + 3.94i)18-s + (2.34 + 2.34i)19-s + ⋯
L(s)  = 1  + (0.933 + 0.359i)2-s + (−0.0359 + 0.0359i)3-s + (0.742 + 0.670i)4-s + (−0.0465 + 0.0206i)6-s + 0.142·7-s + (0.451 + 0.892i)8-s + 0.997i·9-s + (0.713 − 0.713i)11-s + (−0.0508 + 0.00258i)12-s + (−0.489 + 0.489i)13-s + (0.132 + 0.0510i)14-s + (0.101 + 0.994i)16-s − 1.12i·17-s + (−0.358 + 0.930i)18-s + (0.539 + 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.575 - 0.818i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.575 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12265 + 1.10234i\)
\(L(\frac12)\) \(\approx\) \(2.12265 + 1.10234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 - 0.507i)T \)
5 \( 1 \)
good3 \( 1 + (0.0623 - 0.0623i)T - 3iT^{2} \)
7 \( 1 - 0.375T + 7T^{2} \)
11 \( 1 + (-2.36 + 2.36i)T - 11iT^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + 4.64iT - 17T^{2} \)
19 \( 1 + (-2.34 - 2.34i)T + 19iT^{2} \)
23 \( 1 + 2.07T + 23T^{2} \)
29 \( 1 + (2.55 + 2.55i)T + 29iT^{2} \)
31 \( 1 - 8.51T + 31T^{2} \)
37 \( 1 + (7.62 + 7.62i)T + 37iT^{2} \)
41 \( 1 + 3.77iT - 41T^{2} \)
43 \( 1 + (6.21 + 6.21i)T + 43iT^{2} \)
47 \( 1 - 9.71iT - 47T^{2} \)
53 \( 1 + (3.03 + 3.03i)T + 53iT^{2} \)
59 \( 1 + (-8.11 + 8.11i)T - 59iT^{2} \)
61 \( 1 + (-0.728 - 0.728i)T + 61iT^{2} \)
67 \( 1 + (-0.969 + 0.969i)T - 67iT^{2} \)
71 \( 1 + 9.14iT - 71T^{2} \)
73 \( 1 - 7.56T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 3.86iT - 97T^{2} \)
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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

−11.63444221545458213639939572452, −10.80433962027622575990664597163, −9.616862268655148040519453181167, −8.386007908951541204643048551732, −7.54400194353642476650662035010, −6.60719915465666077695156750893, −5.49733479266110402714639248454, −4.67082967958147538696233778338, −3.47602243483081036212233573063, −2.11395361347847457744185057484, 1.46773392371193108831868009738, 3.06925188577219139553043927904, 4.12249796706296241750132082326, 5.16441391998501071146200276723, 6.35583766652160988598066552632, 6.99459295576004747004914956755, 8.356488166333191421527179203133, 9.688163450572069606360023707657, 10.21169245352529701108958495564, 11.49957391294674377252744932219

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