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Properties

Label 2-20e2-20.3-c3-0-20
Degree $2$
Conductor $400$
Sign $0.559 + 0.828i$
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  + (2.35 − 2.35i)3-s + (−0.458 − 0.458i)7-s + 15.9i·9-s − 23.0i·11-s + (−1.65 − 1.65i)13-s + (63.9 − 63.9i)17-s + 65.6·19-s − 2.15·21-s + (45.2 − 45.2i)23-s + (100. + 100. i)27-s + 59.8i·29-s − 279. i·31-s + (−54.2 − 54.2i)33-s + (100. − 100. i)37-s − 7.77·39-s + ⋯
L(s)  = 1  + (0.452 − 0.452i)3-s + (−0.0247 − 0.0247i)7-s + 0.590i·9-s − 0.632i·11-s + (−0.0352 − 0.0352i)13-s + (0.912 − 0.912i)17-s + 0.793·19-s − 0.0224·21-s + (0.410 − 0.410i)23-s + (0.719 + 0.719i)27-s + 0.383i·29-s − 1.61i·31-s + (−0.286 − 0.286i)33-s + (0.446 − 0.446i)37-s − 0.0319·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.559 + 0.828i$
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: \((2,\ 400,\ (\ :3/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.235553074\)
\(L(\frac12)\) \(\approx\) \(2.235553074\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.35 + 2.35i)T - 27iT^{2} \)
7 \( 1 + (0.458 + 0.458i)T + 343iT^{2} \)
11 \( 1 + 23.0iT - 1.33e3T^{2} \)
13 \( 1 + (1.65 + 1.65i)T + 2.19e3iT^{2} \)
17 \( 1 + (-63.9 + 63.9i)T - 4.91e3iT^{2} \)
19 \( 1 - 65.6T + 6.85e3T^{2} \)
23 \( 1 + (-45.2 + 45.2i)T - 1.21e4iT^{2} \)
29 \( 1 - 59.8iT - 2.43e4T^{2} \)
31 \( 1 + 279. iT - 2.97e4T^{2} \)
37 \( 1 + (-100. + 100. i)T - 5.06e4iT^{2} \)
41 \( 1 + 274.T + 6.89e4T^{2} \)
43 \( 1 + (-263. + 263. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-391. - 391. i)T + 1.03e5iT^{2} \)
53 \( 1 + (265. + 265. i)T + 1.48e5iT^{2} \)
59 \( 1 - 121.T + 2.05e5T^{2} \)
61 \( 1 + 617.T + 2.26e5T^{2} \)
67 \( 1 + (-674. - 674. i)T + 3.00e5iT^{2} \)
71 \( 1 + 829. iT - 3.57e5T^{2} \)
73 \( 1 + (-110. - 110. i)T + 3.89e5iT^{2} \)
79 \( 1 + 62.7T + 4.93e5T^{2} \)
83 \( 1 + (-655. + 655. i)T - 5.71e5iT^{2} \)
89 \( 1 + 92.0iT - 7.04e5T^{2} \)
97 \( 1 + (-618. + 618. i)T - 9.12e5iT^{2} \)
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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

−10.76470365887660149915174020575, −9.723489159206760435727031702512, −8.810315047559895941338817968406, −7.81111822560352489141426194350, −7.22841418417708601311366298835, −5.88396028805799669750964870066, −4.91820287061574096265222211537, −3.41793898933906880988970144473, −2.35763920598425276161860708298, −0.822267001719919992232849534752, 1.27311687418621771021707025739, 2.95712439847344203077826395208, 3.89342476927717586765195205512, 5.07123682323733986675034214385, 6.24453183449785098142901813432, 7.33209792116143031061366447846, 8.320298101152083539107578376555, 9.306470953877234526860556117402, 9.927880642176014125380611684588, 10.83338566816038143812548960866

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