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Properties

Label 2-20e2-20.3-c5-0-10
Degree $2$
Conductor $400$
Sign $-0.999 + 0.0299i$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−18.0 + 18.0i)3-s + (25.8 + 25.8i)7-s − 405. i·9-s − 718. i·11-s + (347. + 347. i)13-s + (−954. + 954. i)17-s + 496.·19-s − 931.·21-s + (2.25e3 − 2.25e3i)23-s + (2.92e3 + 2.92e3i)27-s + 5.78e3i·29-s + 5.85e3i·31-s + (1.29e4 + 1.29e4i)33-s + (−9.54e3 + 9.54e3i)37-s − 1.25e4·39-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (0.199 + 0.199i)7-s − 1.66i·9-s − 1.78i·11-s + (0.569 + 0.569i)13-s + (−0.801 + 0.801i)17-s + 0.315·19-s − 0.460·21-s + (0.887 − 0.887i)23-s + (0.773 + 0.773i)27-s + 1.27i·29-s + 1.09i·31-s + (2.06 + 2.06i)33-s + (−1.14 + 1.14i)37-s − 1.31·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0299i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.999 + 0.0299i$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ -0.999 + 0.0299i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.5598295446\)
\(L(\frac12)\) \(\approx\) \(0.5598295446\)
\(L(\frac{7}{2})\) 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 \( 1 + (18.0 - 18.0i)T - 243iT^{2} \)
7 \( 1 + (-25.8 - 25.8i)T + 1.68e4iT^{2} \)
11 \( 1 + 718. iT - 1.61e5T^{2} \)
13 \( 1 + (-347. - 347. i)T + 3.71e5iT^{2} \)
17 \( 1 + (954. - 954. i)T - 1.41e6iT^{2} \)
19 \( 1 - 496.T + 2.47e6T^{2} \)
23 \( 1 + (-2.25e3 + 2.25e3i)T - 6.43e6iT^{2} \)
29 \( 1 - 5.78e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.85e3iT - 2.86e7T^{2} \)
37 \( 1 + (9.54e3 - 9.54e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 6.00e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.19e4 + 1.19e4i)T - 1.47e8iT^{2} \)
47 \( 1 + (-2.44e3 - 2.44e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (-2.27e4 - 2.27e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 6.23e3T + 7.14e8T^{2} \)
61 \( 1 + 4.25e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.41e4 - 2.41e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 2.15e4iT - 1.80e9T^{2} \)
73 \( 1 + (3.67e4 + 3.67e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 9.14e4T + 3.07e9T^{2} \)
83 \( 1 + (7.40e4 - 7.40e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 5.69e4iT - 5.58e9T^{2} \)
97 \( 1 + (-7.17e4 + 7.17e4i)T - 8.58e9iT^{2} \)
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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

−10.77601167018316019922754574876, −10.48886544427019369672288588565, −8.891699524070031281919297835857, −8.697639506599391323620060585875, −6.81237666309300900031018266606, −5.96773464393913106917402439253, −5.19847004340121902526112874386, −4.17319960615871866071594020333, −3.16708624693801736876924589665, −1.11891627387198741455195019564, 0.19169925543589750864633589864, 1.31286577822428940101068665621, 2.37965499250289166974218627814, 4.31289666670684019763831826975, 5.31360432806441932336897836916, 6.23233503947210203020970612430, 7.36104019997168606713678381522, 7.51688845305737602162390111562, 9.141063663293370409695506623126, 10.16916933848484461881593774657

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