Properties

Label 2-20e2-400.189-c1-0-14
Degree $2$
Conductor $400$
Sign $-0.00145 - 0.999i$
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  + (−0.888 − 1.10i)2-s + (1.50 + 2.95i)3-s + (−0.421 + 1.95i)4-s + (−1.10 + 1.94i)5-s + (1.91 − 4.28i)6-s + 3.49·7-s + (2.52 − 1.27i)8-s + (−4.71 + 6.49i)9-s + (3.12 − 0.509i)10-s + (0.229 − 1.44i)11-s + (−6.42 + 1.69i)12-s + (3.88 − 0.615i)13-s + (−3.10 − 3.84i)14-s + (−7.41 − 0.340i)15-s + (−3.64 − 1.64i)16-s + (−1.90 + 0.618i)17-s + ⋯
L(s)  = 1  + (−0.628 − 0.778i)2-s + (0.870 + 1.70i)3-s + (−0.210 + 0.977i)4-s + (−0.494 + 0.869i)5-s + (0.782 − 1.75i)6-s + 1.32·7-s + (0.893 − 0.449i)8-s + (−1.57 + 2.16i)9-s + (0.986 − 0.161i)10-s + (0.0692 − 0.436i)11-s + (−1.85 + 0.490i)12-s + (1.07 − 0.170i)13-s + (−0.830 − 1.02i)14-s + (−1.91 − 0.0880i)15-s + (−0.911 − 0.412i)16-s + (−0.461 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00145 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.00145 - 0.999i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.00145 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934592 + 0.935957i\)
\(L(\frac12)\) \(\approx\) \(0.934592 + 0.935957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.888 + 1.10i)T \)
5 \( 1 + (1.10 - 1.94i)T \)
good3 \( 1 + (-1.50 - 2.95i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 - 3.49T + 7T^{2} \)
11 \( 1 + (-0.229 + 1.44i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-3.88 + 0.615i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.90 - 0.618i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.86 + 3.66i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (3.47 - 2.52i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.14 - 1.09i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-1.82 - 5.63i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.00 + 6.36i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-1.71 + 2.36i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.28 + 7.28i)T + 43iT^{2} \)
47 \( 1 + (-4.08 - 1.32i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.54 + 1.29i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-13.1 + 2.08i)T + (56.1 - 18.2i)T^{2} \)
61 \( 1 + (-13.1 - 2.07i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (-1.99 - 1.01i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (5.62 + 1.82i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.24 + 1.62i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.28 - 3.94i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.29 + 1.67i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-0.888 - 1.22i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.29 + 1.07i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.09561956201888256621955152586, −10.70267632152528235063470884496, −9.837654985238877543811364345283, −8.600421762559402687402886186771, −8.476966219666220577045743401275, −7.32922539606027956125219892139, −5.26371512299221555618587344529, −4.06642410517408331484936976961, −3.49398276408257691545065439332, −2.28279154676226482532918370176, 1.09214551419651604025370279375, 1.94707261221861571853476242175, 4.16785671254821959979511881450, 5.59296223915667035273539716875, 6.64403607860230906370087145864, 7.64073914633896719078647308627, 8.278340776787463489433897888362, 8.525025351727231538624799796093, 9.671625244399064996082557280841, 11.37744764388527982372615958594

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