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Properties

Label 2-20e2-5.4-c3-0-14
Degree $2$
Conductor $400$
Sign $0.894 - 0.447i$
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  + 9i·3-s − 26i·7-s − 54·9-s + 59·11-s − 28i·13-s + 5i·17-s + 109·19-s + 234·21-s − 194i·23-s − 243i·27-s + 32·29-s − 10·31-s + 531i·33-s − 198i·37-s + 252·39-s + ⋯
L(s)  = 1  + 1.73i·3-s − 1.40i·7-s − 2·9-s + 1.61·11-s − 0.597i·13-s + 0.0713i·17-s + 1.31·19-s + 2.43·21-s − 1.75i·23-s − 1.73i·27-s + 0.204·29-s − 0.0579·31-s + 2.80i·33-s − 0.879i·37-s + 1.03·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.983063971\)
\(L(\frac12)\) \(\approx\) \(1.983063971\)
\(L(\frac{5}{2})\) 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 \( 1 - 9iT - 27T^{2} \)
7 \( 1 + 26iT - 343T^{2} \)
11 \( 1 - 59T + 1.33e3T^{2} \)
13 \( 1 + 28iT - 2.19e3T^{2} \)
17 \( 1 - 5iT - 4.91e3T^{2} \)
19 \( 1 - 109T + 6.85e3T^{2} \)
23 \( 1 + 194iT - 1.21e4T^{2} \)
29 \( 1 - 32T + 2.43e4T^{2} \)
31 \( 1 + 10T + 2.97e4T^{2} \)
37 \( 1 + 198iT - 5.06e4T^{2} \)
41 \( 1 - 117T + 6.89e4T^{2} \)
43 \( 1 - 388iT - 7.95e4T^{2} \)
47 \( 1 - 68iT - 1.03e5T^{2} \)
53 \( 1 - 18iT - 1.48e5T^{2} \)
59 \( 1 - 392T + 2.05e5T^{2} \)
61 \( 1 + 710T + 2.26e5T^{2} \)
67 \( 1 - 253iT - 3.00e5T^{2} \)
71 \( 1 - 612T + 3.57e5T^{2} \)
73 \( 1 - 549iT - 3.89e5T^{2} \)
79 \( 1 - 414T + 4.93e5T^{2} \)
83 \( 1 + 121iT - 5.71e5T^{2} \)
89 \( 1 - 81T + 7.04e5T^{2} \)
97 \( 1 + 1.50e3iT - 9.12e5T^{2} \)
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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

−10.79043075621965506230927611353, −9.987603211268030673018956314429, −9.399032736504924727903376186795, −8.407324895568196863055913111781, −7.15146438975704387116974152662, −5.99391113241072707724401966422, −4.67623808152103324851291646450, −4.05934519329582865181703773207, −3.18155000731107341229798617985, −0.812327148724330407414618151455, 1.18780125828249755626139227223, 2.07207612323681275808946182862, 3.40014802688379294242366373557, 5.34276772508495752974998814740, 6.19836486507960859299430145951, 6.96833061995850828091263918969, 7.87219773956820600970951985115, 8.951127617574279610919421900308, 9.434459075668451143500661635437, 11.38656219364851894048553306075

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