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Properties

Label 2-20e2-80.43-c1-0-14
Degree $2$
Conductor $400$
Sign $-0.474 - 0.879i$
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  + (1.34 + 0.430i)2-s + 2.96i·3-s + (1.62 + 1.15i)4-s + (−1.27 + 3.99i)6-s + (0.115 − 0.115i)7-s + (1.69 + 2.26i)8-s − 5.79·9-s + (2.95 − 2.95i)11-s + (−3.43 + 4.83i)12-s − 1.55·13-s + (0.204 − 0.105i)14-s + (1.31 + 3.77i)16-s + (−0.299 + 0.299i)17-s + (−7.80 − 2.49i)18-s + (2.26 − 2.26i)19-s + ⋯
L(s)  = 1  + (0.952 + 0.304i)2-s + 1.71i·3-s + (0.814 + 0.579i)4-s + (−0.520 + 1.63i)6-s + (0.0435 − 0.0435i)7-s + (0.599 + 0.800i)8-s − 1.93·9-s + (0.892 − 0.892i)11-s + (−0.992 + 1.39i)12-s − 0.432·13-s + (0.0546 − 0.0282i)14-s + (0.327 + 0.944i)16-s + (−0.0726 + 0.0726i)17-s + (−1.84 − 0.587i)18-s + (0.519 − 0.519i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.474 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.474 - 0.879i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.474 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22691 + 2.05650i\)
\(L(\frac12)\) \(\approx\) \(1.22691 + 2.05650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.430i)T \)
5 \( 1 \)
good3 \( 1 - 2.96iT - 3T^{2} \)
7 \( 1 + (-0.115 + 0.115i)T - 7iT^{2} \)
11 \( 1 + (-2.95 + 2.95i)T - 11iT^{2} \)
13 \( 1 + 1.55T + 13T^{2} \)
17 \( 1 + (0.299 - 0.299i)T - 17iT^{2} \)
19 \( 1 + (-2.26 + 2.26i)T - 19iT^{2} \)
23 \( 1 + (4.14 + 4.14i)T + 23iT^{2} \)
29 \( 1 + (0.289 + 0.289i)T + 29iT^{2} \)
31 \( 1 + 4.18iT - 31T^{2} \)
37 \( 1 + 1.63T + 37T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 - 6.72T + 43T^{2} \)
47 \( 1 + (-4.38 - 4.38i)T + 47iT^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (1.63 + 1.63i)T + 59iT^{2} \)
61 \( 1 + (1.23 - 1.23i)T - 61iT^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 + (1.12 - 1.12i)T - 73iT^{2} \)
79 \( 1 + 3.62T + 79T^{2} \)
83 \( 1 + 1.62iT - 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + (9.69 - 9.69i)T - 97iT^{2} \)
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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

−11.41290552816336335745880224817, −10.88048957900888919965109640625, −9.821753319121606679045439569618, −8.940910582760840022325049540229, −7.929825323966673525918865398209, −6.46162090435292901824533405569, −5.58645167326469695520423584593, −4.56076726874743837065415704361, −3.86491188177027303869584975385, −2.80595641879891617840256796437, 1.39883872016487744784952507373, 2.35177492387264684305352175992, 3.82468283200205258145687636036, 5.31214372929085080343140185364, 6.25678943454640997376539993096, 7.13592434584353293483582714410, 7.66703063798265994520581071040, 9.129782145645469657767078734255, 10.30360389240522955274968474159, 11.53737508478773994848368242970

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