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Properties

Label 2-20e2-25.6-c1-0-8
Degree $2$
Conductor $400$
Sign $-0.535 + 0.844i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 1.31i)3-s + (−0.690 + 2.12i)5-s − 2.61·7-s + (0.618 − 1.90i)9-s + (−0.381 − 1.17i)11-s + (1.04 − 3.21i)13-s + (−1.54 − 4.75i)15-s + (−4.23 − 3.07i)17-s + (6.16 + 4.47i)19-s + (4.73 − 3.44i)21-s + (−1.69 − 5.20i)23-s + (−4.04 − 2.93i)25-s + (−0.690 − 2.12i)27-s + (−3.54 + 2.57i)29-s + (−3.80 − 2.76i)31-s + ⋯
L(s)  = 1  + (−1.04 + 0.758i)3-s + (−0.309 + 0.951i)5-s − 0.989·7-s + (0.206 − 0.634i)9-s + (−0.115 − 0.354i)11-s + (0.289 − 0.892i)13-s + (−0.398 − 1.22i)15-s + (−1.02 − 0.746i)17-s + (1.41 + 1.02i)19-s + (1.03 − 0.750i)21-s + (−0.352 − 1.08i)23-s + (−0.809 − 0.587i)25-s + (−0.132 − 0.409i)27-s + (−0.658 + 0.478i)29-s + (−0.684 − 0.497i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.535 + 0.844i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.31i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 + (0.381 + 1.17i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.04 + 3.21i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.23 + 3.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.16 - 4.47i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.69 + 5.20i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.54 - 2.57i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.80 + 2.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.54 - 7.83i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.76 + 5.42i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.38T + 43T^{2} \)
47 \( 1 + (-0.118 + 0.0857i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.42 - 4.66i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.736 - 2.26i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.07 + 12.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.0 - 8.05i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.97 - 5.06i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.39 - 10.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.30 - 3.13i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.66 + 2.66i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.23 - 3.80i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.3 + 8.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.93414523814018423415633654728, −10.20641412460064597762105154549, −9.555572214315036933624235552686, −8.121635684160960241958886040545, −6.96369532015327145234516389826, −6.10180114193618645303193275658, −5.27166738843857013154640463296, −3.88322645835256626350665122270, −2.92643952965495895935227326127, 0, 1.62200072195613404893689762798, 3.67674313916135978279551366416, 4.96039199531407040063262256321, 5.92953992121821551683549367961, 6.81336460077360910137536913667, 7.61938043585602138634836534640, 9.047252826338054966984326176819, 9.554151883967883297101270097131, 11.05505474913033768898092782120

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