Properties

Label 2-20e2-400.123-c1-0-52
Degree $2$
Conductor $400$
Sign $0.389 + 0.921i$
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  + (0.608 − 1.27i)2-s + (3.11 + 1.01i)3-s + (−1.25 − 1.55i)4-s + (0.0755 − 2.23i)5-s + (3.18 − 3.35i)6-s + (−1.39 + 1.39i)7-s + (−2.74 + 0.662i)8-s + (6.23 + 4.53i)9-s + (−2.80 − 1.45i)10-s + (0.459 − 2.90i)11-s + (−2.34 − 6.10i)12-s + (1.35 + 0.982i)13-s + (0.933 + 2.63i)14-s + (2.49 − 6.87i)15-s + (−0.826 + 3.91i)16-s + (−3.30 − 1.68i)17-s + ⋯
L(s)  = 1  + (0.430 − 0.902i)2-s + (1.79 + 0.583i)3-s + (−0.629 − 0.776i)4-s + (0.0338 − 0.999i)5-s + (1.30 − 1.37i)6-s + (−0.527 + 0.527i)7-s + (−0.972 + 0.234i)8-s + (2.07 + 1.51i)9-s + (−0.887 − 0.460i)10-s + (0.138 − 0.875i)11-s + (−0.678 − 1.76i)12-s + (0.374 + 0.272i)13-s + (0.249 + 0.703i)14-s + (0.644 − 1.77i)15-s + (−0.206 + 0.978i)16-s + (−0.801 − 0.408i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.389 + 0.921i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.389 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19374 - 1.45480i\)
\(L(\frac12)\) \(\approx\) \(2.19374 - 1.45480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 + 1.27i)T \)
5 \( 1 + (-0.0755 + 2.23i)T \)
good3 \( 1 + (-3.11 - 1.01i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.39 - 1.39i)T - 7iT^{2} \)
11 \( 1 + (-0.459 + 2.90i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-1.35 - 0.982i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.30 + 1.68i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-5.31 - 2.70i)T + (11.1 + 15.3i)T^{2} \)
23 \( 1 + (6.20 + 0.982i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (6.23 - 3.17i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-3.15 + 1.02i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.86 - 2.81i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.18 - 5.75i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.66T + 43T^{2} \)
47 \( 1 + (2.15 - 1.10i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (5.64 + 1.83i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.758 - 4.79i)T + (-56.1 + 18.2i)T^{2} \)
61 \( 1 + (-0.761 + 4.80i)T + (-58.0 - 18.8i)T^{2} \)
67 \( 1 + (3.61 + 11.1i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.49 + 13.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.269 - 1.70i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (1.58 - 4.87i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.37 - 2.07i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.60 - 3.34i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.775 + 1.52i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.09189890137601305063048898140, −9.754969725463475410699452765797, −9.442805587608391577899893083853, −8.674870001452353524542470676476, −7.952515040446201300561728657307, −6.02013202702600915676174376006, −4.77033236509624099110939277062, −3.81015380631241723685620097936, −2.97544891419133220801107010860, −1.71330888268911336816191635734, 2.31975348852856750131839120511, 3.44579412785643957274039844985, 4.13121838804907326019418080264, 6.12753094096781424967238528145, 7.14342639418476627970708793668, 7.42104409393087569773659671405, 8.418230095705659820072046602503, 9.458160778907504844526187804699, 10.04277466160464229114688656026, 11.72539542400593636922652987402

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