Properties

Label 2-20e2-400.221-c1-0-12
Degree $2$
Conductor $400$
Sign $-0.830 - 0.556i$
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  + (0.368 + 1.36i)2-s + (0.0334 + 0.211i)3-s + (−1.72 + 1.00i)4-s + (1.50 + 1.64i)5-s + (−0.276 + 0.123i)6-s + 3.25i·7-s + (−2.00 − 1.98i)8-s + (2.80 − 0.912i)9-s + (−1.69 + 2.66i)10-s + (−1.61 − 3.16i)11-s + (−0.270 − 0.331i)12-s + (−2.64 + 5.19i)13-s + (−4.44 + 1.19i)14-s + (−0.298 + 0.374i)15-s + (1.97 − 3.47i)16-s + (2.01 − 1.46i)17-s + ⋯
L(s)  = 1  + (0.260 + 0.965i)2-s + (0.0193 + 0.121i)3-s + (−0.864 + 0.502i)4-s + (0.675 + 0.737i)5-s + (−0.112 + 0.0504i)6-s + 1.22i·7-s + (−0.710 − 0.703i)8-s + (0.936 − 0.304i)9-s + (−0.536 + 0.843i)10-s + (−0.485 − 0.953i)11-s + (−0.0780 − 0.0957i)12-s + (−0.733 + 1.43i)13-s + (−1.18 + 0.320i)14-s + (−0.0769 + 0.0966i)15-s + (0.494 − 0.869i)16-s + (0.489 − 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.830 - 0.556i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442401 + 1.45602i\)
\(L(\frac12)\) \(\approx\) \(0.442401 + 1.45602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.368 - 1.36i)T \)
5 \( 1 + (-1.50 - 1.64i)T \)
good3 \( 1 + (-0.0334 - 0.211i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 - 3.25iT - 7T^{2} \)
11 \( 1 + (1.61 + 3.16i)T + (-6.46 + 8.89i)T^{2} \)
13 \( 1 + (2.64 - 5.19i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-2.01 + 1.46i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.30 - 0.365i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (6.76 + 2.19i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.560 - 3.53i)T + (-27.5 + 8.96i)T^{2} \)
31 \( 1 + (-0.377 + 0.273i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.95 + 3.54i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-8.02 + 2.60i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.97 + 1.97i)T - 43iT^{2} \)
47 \( 1 + (-7.34 - 5.33i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.52 + 1.35i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (6.59 + 3.36i)T + (34.6 + 47.7i)T^{2} \)
61 \( 1 + (-6.49 + 3.31i)T + (35.8 - 49.3i)T^{2} \)
67 \( 1 + (3.70 + 0.586i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-5.25 + 7.23i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.99 - 2.27i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.38 + 2.45i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-17.0 - 2.70i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.24 - 1.05i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.52 - 1.10i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.96543681491552876749076474501, −10.54841904963011430438680191287, −9.480423358723103119454751350163, −9.047751492962623297804712706878, −7.73861200794804107242194002988, −6.81829224255125017598180586584, −5.96505248021470852957057729052, −5.15853348299754355665471822591, −3.77069596870693512684693563913, −2.39057806799282111458930361566, 0.990524490079922972442149813913, 2.28338786093371645696576829164, 3.93225821394509964033238903468, 4.81956341823673492959709289041, 5.71548747689085678934788677006, 7.37444536136341478918531926246, 8.095785900492487125065985168304, 9.643837225534971608593462700131, 10.14026471457964919146679491817, 10.51918614196847040668013738118

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