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Properties

Label 2-20e2-25.16-c1-0-4
Degree $2$
Conductor $400$
Sign $-0.187 - 0.982i$
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 + 2.48i)3-s + (0.690 + 2.12i)5-s + 3·7-s + (−3.11 + 2.26i)9-s + (−0.190 − 0.138i)11-s + (−0.809 + 0.587i)13-s + (−4.73 + 3.44i)15-s + (2.42 − 7.46i)17-s + (0.263 − 0.812i)19-s + (2.42 + 7.46i)21-s + (−5.04 − 3.66i)23-s + (−4.04 + 2.93i)25-s + (−1.80 − 1.31i)27-s + (−0.163 − 0.502i)29-s + (−1.30 + 4.02i)31-s + ⋯
L(s)  = 1  + (0.467 + 1.43i)3-s + (0.309 + 0.951i)5-s + 1.13·7-s + (−1.03 + 0.755i)9-s + (−0.0575 − 0.0418i)11-s + (−0.224 + 0.163i)13-s + (−1.22 + 0.888i)15-s + (0.588 − 1.81i)17-s + (0.0605 − 0.186i)19-s + (0.529 + 1.63i)21-s + (−1.05 − 0.764i)23-s + (−0.809 + 0.587i)25-s + (−0.348 − 0.252i)27-s + (−0.0302 − 0.0932i)29-s + (−0.235 + 0.723i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14092 + 1.37914i\)
\(L(\frac12)\) \(\approx\) \(1.14092 + 1.37914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 2.48i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + (0.190 + 0.138i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.42 + 7.46i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.263 + 0.812i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.04 + 3.66i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.163 + 0.502i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.30 - 4.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.92 - 4.30i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.04 + 4.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.76T + 43T^{2} \)
47 \( 1 + (1.83 + 5.65i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.472 - 1.45i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.61 + 2.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.73 - 1.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.78 + 5.48i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.927 - 2.85i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.61 - 3.35i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.854 - 2.62i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.16 + 12.8i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.61 - 2.62i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.26 + 10.0i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.27072811477159926282063673141, −10.48530255579814108964512538484, −9.839870178187994494753068240690, −8.992802385211804908178235465826, −7.952299237160884627087736734550, −6.93521912190920406055813921907, −5.42161843309479855392362108319, −4.65785322453635944543271097618, −3.47526936478783636205290516813, −2.37283253929109365254223604174, 1.33740406821120752643164221842, 2.08217233944718027530155144274, 3.99851397107739463628333312448, 5.39413828176502001811686899506, 6.23180748413829489444945278014, 7.75269501983470168296675497588, 7.961595600537195664352958952845, 8.849529062651119350805966182382, 10.03298960002922111665472756603, 11.25497820441579356448209688490

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