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Properties

Label 2-20e2-20.7-c3-0-5
Degree $2$
Conductor $400$
Sign $0.880 - 0.473i$
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  + (−2.99 − 2.99i)3-s + (6.68 − 6.68i)7-s − 9.09i·9-s + 63.1i·11-s + (−28.2 + 28.2i)13-s + (−30.3 − 30.3i)17-s + 22.2·19-s − 40.0·21-s + (86.5 + 86.5i)23-s + (−107. + 107. i)27-s − 188. i·29-s − 71.2i·31-s + (188. − 188. i)33-s + (217. + 217. i)37-s + 168.·39-s + ⋯
L(s)  = 1  + (−0.575 − 0.575i)3-s + (0.361 − 0.361i)7-s − 0.336i·9-s + 1.73i·11-s + (−0.602 + 0.602i)13-s + (−0.433 − 0.433i)17-s + 0.268·19-s − 0.415·21-s + (0.784 + 0.784i)23-s + (−0.769 + 0.769i)27-s − 1.20i·29-s − 0.413i·31-s + (0.996 − 0.996i)33-s + (0.964 + 0.964i)37-s + 0.693·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.318063633\)
\(L(\frac12)\) \(\approx\) \(1.318063633\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.99 + 2.99i)T + 27iT^{2} \)
7 \( 1 + (-6.68 + 6.68i)T - 343iT^{2} \)
11 \( 1 - 63.1iT - 1.33e3T^{2} \)
13 \( 1 + (28.2 - 28.2i)T - 2.19e3iT^{2} \)
17 \( 1 + (30.3 + 30.3i)T + 4.91e3iT^{2} \)
19 \( 1 - 22.2T + 6.85e3T^{2} \)
23 \( 1 + (-86.5 - 86.5i)T + 1.21e4iT^{2} \)
29 \( 1 + 188. iT - 2.43e4T^{2} \)
31 \( 1 + 71.2iT - 2.97e4T^{2} \)
37 \( 1 + (-217. - 217. i)T + 5.06e4iT^{2} \)
41 \( 1 + 4.57T + 6.89e4T^{2} \)
43 \( 1 + (-392. - 392. i)T + 7.95e4iT^{2} \)
47 \( 1 + (280. - 280. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-82.3 + 82.3i)T - 1.48e5iT^{2} \)
59 \( 1 - 794.T + 2.05e5T^{2} \)
61 \( 1 - 714.T + 2.26e5T^{2} \)
67 \( 1 + (-373. + 373. i)T - 3.00e5iT^{2} \)
71 \( 1 - 528. iT - 3.57e5T^{2} \)
73 \( 1 + (-531. + 531. i)T - 3.89e5iT^{2} \)
79 \( 1 + 349.T + 4.93e5T^{2} \)
83 \( 1 + (-112. - 112. i)T + 5.71e5iT^{2} \)
89 \( 1 + 424. iT - 7.04e5T^{2} \)
97 \( 1 + (501. + 501. i)T + 9.12e5iT^{2} \)
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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

−11.28029510031533832678395674715, −9.788146008405428812842714074200, −9.430698802162173229527846855854, −7.80106818225563519239630346585, −7.15337756203982981649716660407, −6.36502129284057763690112419594, −5.02776292652152002668864732555, −4.18138792932687291488371482227, −2.36630351252272570435169948379, −1.07327536206683353925103534015, 0.57525860402511737446073501773, 2.50270169543341845881814358070, 3.83094657255593621080382250453, 5.19278064360904661654743561138, 5.62524950371248648185549050737, 6.93335114427589800256865515918, 8.229777221881297331462095969995, 8.832745098504614378296036253465, 10.10854258770323050739733462813, 10.92469615219343023655939263719

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