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Properties

Label 2-20e2-80.67-c1-0-30
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: $\chi_{400} (307, \cdot )$
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

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

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