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Properties

Label 2-20e2-100.27-c1-0-5
Degree $2$
Conductor $400$
Sign $0.770 - 0.637i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 + 0.510i)3-s + (−0.844 − 2.07i)5-s + (−0.941 + 0.941i)7-s + (−1.02 + 1.40i)9-s + (3.06 + 4.21i)11-s + (6.49 − 1.02i)13-s + (1.90 + 1.64i)15-s + (0.833 − 1.63i)17-s + (0.753 + 2.31i)19-s + (0.462 − 1.42i)21-s + (7.19 + 1.13i)23-s + (−3.57 + 3.49i)25-s + (0.832 − 5.25i)27-s + (5.17 + 1.68i)29-s + (−9.74 + 3.16i)31-s + ⋯
L(s)  = 1  + (−0.578 + 0.294i)3-s + (−0.377 − 0.925i)5-s + (−0.355 + 0.355i)7-s + (−0.340 + 0.468i)9-s + (0.923 + 1.27i)11-s + (1.80 − 0.285i)13-s + (0.491 + 0.424i)15-s + (0.202 − 0.396i)17-s + (0.172 + 0.532i)19-s + (0.100 − 0.310i)21-s + (1.49 + 0.237i)23-s + (−0.714 + 0.699i)25-s + (0.160 − 1.01i)27-s + (0.960 + 0.312i)29-s + (−1.75 + 0.568i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.770 - 0.637i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00614 + 0.362016i\)
\(L(\frac12)\) \(\approx\) \(1.00614 + 0.362016i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.844 + 2.07i)T \)
good3 \( 1 + (1.00 - 0.510i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (0.941 - 0.941i)T - 7iT^{2} \)
11 \( 1 + (-3.06 - 4.21i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-6.49 + 1.02i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.833 + 1.63i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-0.753 - 2.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.19 - 1.13i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-5.17 - 1.68i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.74 - 3.16i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.379 - 2.39i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.485 + 0.352i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.84 - 4.84i)T + 43iT^{2} \)
47 \( 1 + (2.93 + 5.76i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.53 - 3.01i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (8.77 + 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.27 + 1.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (6.68 + 3.40i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-6.56 - 2.13i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.502 + 3.16i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-2.00 + 6.15i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.97 + 3.88i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-4.09 - 5.62i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.6 - 5.43i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.38408662926258771160076622329, −10.63165730743454182079699198010, −9.387329275303513949421571085491, −8.814611858390589218253299643952, −7.73320795156834772039982486723, −6.50265254662981537285216350671, −5.46958053107754126972589296638, −4.62862395078611533524488773135, −3.44839283104202116168243322519, −1.38332392443889829462650975415, 0.913054328340596064107694649726, 3.22292434454072183759952226863, 3.86894606356769289745010435039, 5.80771895429270152670420049599, 6.39677862859868664843314410909, 7.12475230030936632907126872280, 8.527047790586858961906753610831, 9.202405230171199258451319351282, 10.77239369081197015463621486192, 11.09763406189677369250457336467

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