Properties

Label 2-20e2-400.181-c1-0-51
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: $\chi_{400} (181, \cdot )$
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

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

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