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Properties

Label 2-20e2-20.3-c3-0-26
Degree $2$
Conductor $400$
Sign $-0.997 + 0.0706i$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.59 − 6.59i)3-s + (−17.4 − 17.4i)7-s − 59.9i·9-s + 1.16i·11-s + (−32.6 − 32.6i)13-s + (−90.9 + 90.9i)17-s + 131.·19-s − 229.·21-s + (−98.9 + 98.9i)23-s + (−217. − 217. i)27-s − 167. i·29-s + 60.1i·31-s + (7.71 + 7.71i)33-s + (193. − 193. i)37-s − 430.·39-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s + (−0.941 − 0.941i)7-s − 2.22i·9-s + 0.0320i·11-s + (−0.696 − 0.696i)13-s + (−1.29 + 1.29i)17-s + 1.58·19-s − 2.38·21-s + (−0.897 + 0.897i)23-s + (−1.54 − 1.54i)27-s − 1.07i·29-s + 0.348i·31-s + (0.0406 + 0.0406i)33-s + (0.859 − 0.859i)37-s − 1.76·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.997 + 0.0706i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ -0.997 + 0.0706i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.802337800\)
\(L(\frac12)\) \(\approx\) \(1.802337800\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3 \( 1 + (-6.59 + 6.59i)T - 27iT^{2} \)
7 \( 1 + (17.4 + 17.4i)T + 343iT^{2} \)
11 \( 1 - 1.16iT - 1.33e3T^{2} \)
13 \( 1 + (32.6 + 32.6i)T + 2.19e3iT^{2} \)
17 \( 1 + (90.9 - 90.9i)T - 4.91e3iT^{2} \)
19 \( 1 - 131.T + 6.85e3T^{2} \)
23 \( 1 + (98.9 - 98.9i)T - 1.21e4iT^{2} \)
29 \( 1 + 167. iT - 2.43e4T^{2} \)
31 \( 1 - 60.1iT - 2.97e4T^{2} \)
37 \( 1 + (-193. + 193. i)T - 5.06e4iT^{2} \)
41 \( 1 - 28.7T + 6.89e4T^{2} \)
43 \( 1 + (-76.6 + 76.6i)T - 7.95e4iT^{2} \)
47 \( 1 + (176. + 176. i)T + 1.03e5iT^{2} \)
53 \( 1 + (327. + 327. i)T + 1.48e5iT^{2} \)
59 \( 1 - 141.T + 2.05e5T^{2} \)
61 \( 1 - 369.T + 2.26e5T^{2} \)
67 \( 1 + (-67.8 - 67.8i)T + 3.00e5iT^{2} \)
71 \( 1 + 46.7iT - 3.57e5T^{2} \)
73 \( 1 + (-141. - 141. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.03e3T + 4.93e5T^{2} \)
83 \( 1 + (-498. + 498. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.53e3iT - 7.04e5T^{2} \)
97 \( 1 + (1.20 - 1.20i)T - 9.12e5iT^{2} \)
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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

−10.08348746188641124117509816235, −9.480718028812613641308927043989, −8.333910699416726171764187725289, −7.57387053519657637155649609761, −6.93651526024703157531445521275, −5.93707992099467121421854206336, −3.97777359571307356220570967164, −3.09314539331514450679861530567, −1.89470722733848088682460785841, −0.47753897841414488553544882585, 2.44917552105375092220994586503, 3.03144160203693231859556739043, 4.32147215299080993724362621057, 5.19016842886119302402683585690, 6.64657916294049540387242802639, 7.84177895032708379232204941587, 8.962872619063736021308676983288, 9.411348311806986119318865359039, 9.924702737770019727248920041684, 11.17379179429748893888960149783

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