Properties

Label 2-20e2-400.227-c1-0-12
Degree $2$
Conductor $400$
Sign $-0.590 + 0.807i$
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  + (−0.839 − 1.13i)2-s + (−3.16 − 1.02i)3-s + (−0.589 + 1.91i)4-s + (0.0858 + 2.23i)5-s + (1.48 + 4.46i)6-s + (−0.838 − 0.838i)7-s + (2.66 − 0.934i)8-s + (6.51 + 4.73i)9-s + (2.47 − 1.97i)10-s + (−1.86 − 0.295i)11-s + (3.82 − 5.43i)12-s + (0.377 + 0.274i)13-s + (−0.249 + 1.65i)14-s + (2.02 − 7.15i)15-s + (−3.30 − 2.25i)16-s + (2.08 − 4.08i)17-s + ⋯
L(s)  = 1  + (−0.593 − 0.804i)2-s + (−1.82 − 0.593i)3-s + (−0.294 + 0.955i)4-s + (0.0384 + 0.999i)5-s + (0.606 + 1.82i)6-s + (−0.316 − 0.316i)7-s + (0.943 − 0.330i)8-s + (2.17 + 1.57i)9-s + (0.781 − 0.624i)10-s + (−0.563 − 0.0892i)11-s + (1.10 − 1.56i)12-s + (0.104 + 0.0760i)13-s + (−0.0667 + 0.443i)14-s + (0.522 − 1.84i)15-s + (−0.826 − 0.563i)16-s + (0.504 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.590 + 0.807i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.590 + 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158403 - 0.312041i\)
\(L(\frac12)\) \(\approx\) \(0.158403 - 0.312041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.839 + 1.13i)T \)
5 \( 1 + (-0.0858 - 2.23i)T \)
good3 \( 1 + (3.16 + 1.02i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.838 + 0.838i)T + 7iT^{2} \)
11 \( 1 + (1.86 + 0.295i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (-0.377 - 0.274i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.08 + 4.08i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.912 - 1.79i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (0.627 - 3.96i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (3.25 + 6.38i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (-5.55 + 1.80i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.92 + 2.12i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.14 + 4.32i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (5.11 + 10.0i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (11.6 + 3.79i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.89 + 1.25i)T + (56.1 - 18.2i)T^{2} \)
61 \( 1 + (-6.71 - 1.06i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (0.330 + 1.01i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.28 + 10.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.93 - 1.25i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.203 + 0.625i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.57 - 1.16i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.478 - 0.347i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-10.7 + 5.50i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.14279793280051047221985337758, −10.21577476573885080382931581169, −9.773751414078950144416254710655, −7.82601552503944015374582016160, −7.25124867117704797772325981873, −6.30981353029002442944644475810, −5.22937157877928017330364635537, −3.80510716775633348970824949660, −2.16410284801520180302075317673, −0.43222391714228316238880530152, 1.10025821940614343140064126949, 4.32022002988305591867496014294, 5.08838637730485897330036535835, 5.87160583016564294098919676554, 6.54719221542531755602954443744, 7.86145048098838722432768732509, 8.997352674838842878282903330465, 9.828880151605020727178983800713, 10.56855866649949956895494495289, 11.31952165585663048344672508787

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