Properties

Label 2-20e2-25.14-c1-0-5
Degree $2$
Conductor $400$
Sign $0.536 - 0.843i$
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  + (2.47 + 0.804i)3-s + (−1.07 + 1.95i)5-s + 0.407i·7-s + (3.05 + 2.21i)9-s + (1.61 − 1.17i)11-s + (−0.411 + 0.566i)13-s + (−4.24 + 3.98i)15-s + (1.50 − 0.489i)17-s + (1.52 + 4.70i)19-s + (−0.327 + 1.00i)21-s + (0.706 + 0.971i)23-s + (−2.67 − 4.22i)25-s + (1.18 + 1.63i)27-s + (−1.70 + 5.23i)29-s + (−2.53 − 7.80i)31-s + ⋯
L(s)  = 1  + (1.42 + 0.464i)3-s + (−0.482 + 0.876i)5-s + 0.153i·7-s + (1.01 + 0.739i)9-s + (0.487 − 0.354i)11-s + (−0.114 + 0.157i)13-s + (−1.09 + 1.02i)15-s + (0.365 − 0.118i)17-s + (0.350 + 1.08i)19-s + (−0.0714 + 0.219i)21-s + (0.147 + 0.202i)23-s + (−0.534 − 0.844i)25-s + (0.228 + 0.313i)27-s + (−0.316 + 0.972i)29-s + (−0.455 − 1.40i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.536 - 0.843i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.536 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75660 + 0.964555i\)
\(L(\frac12)\) \(\approx\) \(1.75660 + 0.964555i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 - 1.95i)T \)
good3 \( 1 + (-2.47 - 0.804i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.407iT - 7T^{2} \)
11 \( 1 + (-1.61 + 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.411 - 0.566i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.50 + 0.489i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.52 - 4.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.706 - 0.971i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.70 - 5.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.53 + 7.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.01 + 4.15i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.83 + 4.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + (1.21 + 0.393i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.83 + 1.56i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.25 - 3.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.62 + 5.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.93 + 0.952i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.12 - 6.53i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.320 + 0.441i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.69 + 5.21i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.926 + 0.301i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.83 - 1.33i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (14.4 + 4.70i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.34087691233992897248957555504, −10.32361751072188238435129917756, −9.542298114810583990221801500487, −8.674517199838055686874023453079, −7.82384010118744385314379078859, −7.02929616880067504753258502570, −5.65318666503409981835759958540, −3.94579528165030752393917880870, −3.43066165702376776945838166494, −2.20594697988415224021044429765, 1.35241333004086309481427961335, 2.84472036206462872095673546227, 3.97073037847156554812408261758, 5.07195894134094880564109653869, 6.73462854014285557305019401723, 7.65656653406963873748289697971, 8.352118460972520895125078894923, 9.132555128537338622425995012084, 9.833672163375603566983210829716, 11.29611515205987406047191842358

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