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Properties

Label 2-20e2-25.14-c1-0-9
Degree $2$
Conductor $400$
Sign $0.711 + 0.702i$
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  + (0.811 + 0.263i)3-s + (−0.390 − 2.20i)5-s + 1.47i·7-s + (−1.83 − 1.33i)9-s + (3.08 − 2.24i)11-s + (3.25 − 4.47i)13-s + (0.263 − 1.88i)15-s + (4.28 − 1.39i)17-s + (2.12 + 6.55i)19-s + (−0.389 + 1.19i)21-s + (−2.45 − 3.38i)23-s + (−4.69 + 1.72i)25-s + (−2.64 − 3.63i)27-s + (−0.819 + 2.52i)29-s + (−1.82 − 5.61i)31-s + ⋯
L(s)  = 1  + (0.468 + 0.152i)3-s + (−0.174 − 0.984i)5-s + 0.558i·7-s + (−0.612 − 0.445i)9-s + (0.929 − 0.675i)11-s + (0.902 − 1.24i)13-s + (0.0679 − 0.487i)15-s + (1.03 − 0.337i)17-s + (0.488 + 1.50i)19-s + (−0.0850 + 0.261i)21-s + (−0.512 − 0.704i)23-s + (−0.938 + 0.344i)25-s + (−0.508 − 0.700i)27-s + (−0.152 + 0.468i)29-s + (−0.327 − 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44565 - 0.593895i\)
\(L(\frac12)\) \(\approx\) \(1.44565 - 0.593895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.390 + 2.20i)T \)
good3 \( 1 + (-0.811 - 0.263i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 1.47iT - 7T^{2} \)
11 \( 1 + (-3.08 + 2.24i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.25 + 4.47i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.28 + 1.39i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.12 - 6.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.45 + 3.38i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.819 - 2.52i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.82 + 5.61i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.29 - 3.16i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.71 - 4.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.36iT - 43T^{2} \)
47 \( 1 + (2.89 + 0.940i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.514 + 0.167i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.07 - 2.96i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.605 + 0.439i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (12.6 - 4.11i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.19 + 3.66i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.63 + 3.62i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.13 - 15.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.71 + 0.557i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (10.4 - 7.61i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (7.33 + 2.38i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.37676007382992322452373104977, −10.02825381811750265098883591624, −9.202878680837976945252746075497, −8.382577150030893578117941903288, −7.916275426133166209148365365839, −5.99489644948657022229735138248, −5.62103464931407706841801531480, −3.96842039950080652963691689378, −3.11698684648778735115065652417, −1.12067280991398383076511514257, 1.84486443089855156490537151877, 3.28337089137928805244268718184, 4.18931300686925905726494037887, 5.77914991043071356232575332183, 6.96315401603240482472529021265, 7.43073671082761152150536823175, 8.697101176946989979052460950797, 9.494799246407197555581352516745, 10.59136804284263844731205704954, 11.36182611036597457983274567346

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