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Properties

Label 2-20e2-16.13-c1-0-30
Degree $2$
Conductor $400$
Sign $-0.880 + 0.474i$
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  + (0.507 − 1.31i)2-s + (−0.0623 − 0.0623i)3-s + (−1.48 − 1.34i)4-s + (−0.113 + 0.0506i)6-s − 0.375i·7-s + (−2.52 + 1.27i)8-s − 2.99i·9-s + (2.36 − 2.36i)11-s + (0.00895 + 0.176i)12-s + (−1.76 − 1.76i)13-s + (−0.496 − 0.190i)14-s + (0.405 + 3.97i)16-s − 4.64·17-s + (−3.94 − 1.51i)18-s + (−2.34 − 2.34i)19-s + ⋯
L(s)  = 1  + (0.359 − 0.933i)2-s + (−0.0359 − 0.0359i)3-s + (−0.742 − 0.670i)4-s + (−0.0465 + 0.0206i)6-s − 0.142i·7-s + (−0.892 + 0.451i)8-s − 0.997i·9-s + (0.713 − 0.713i)11-s + (0.00258 + 0.0508i)12-s + (−0.489 − 0.489i)13-s + (−0.132 − 0.0510i)14-s + (0.101 + 0.994i)16-s − 1.12·17-s + (−0.930 − 0.358i)18-s + (−0.539 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.880 + 0.474i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312238 - 1.23745i\)
\(L(\frac12)\) \(\approx\) \(0.312238 - 1.23745i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.507 + 1.31i)T \)
5 \( 1 \)
good3 \( 1 + (0.0623 + 0.0623i)T + 3iT^{2} \)
7 \( 1 + 0.375iT - 7T^{2} \)
11 \( 1 + (-2.36 + 2.36i)T - 11iT^{2} \)
13 \( 1 + (1.76 + 1.76i)T + 13iT^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
19 \( 1 + (2.34 + 2.34i)T + 19iT^{2} \)
23 \( 1 + 2.07iT - 23T^{2} \)
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 \( 1 - 9.71T + 47T^{2} \)
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 \( 1 - 7.56iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + 15.7iT - 89T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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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

−10.89616727538230538803610403064, −10.23638348530759459636448495031, −9.049805932322025028802604168721, −8.622921414082732471755844828359, −6.85738481035133893800899453876, −6.03558225983380729336569813615, −4.72376377017053055115091809459, −3.74253333062109631722913662565, −2.57292286252741526345611972687, −0.76788347845178605635005247704, 2.32507624084420417071665906256, 4.13406268450796211920296872927, 4.79185783261197339484212699211, 6.02849861099183347136540673286, 6.93878734564095676027641738830, 7.79430635712433252908647199841, 8.779875153432651289225268130938, 9.604647011380885201127452088070, 10.72510747152727973325776768293, 11.92070894332425069119485982319

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