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Properties

Label 2-20e2-100.27-c1-0-8
Degree $2$
Conductor $400$
Sign $0.999 + 0.0404i$
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  + (0.192 − 0.0980i)3-s + (1.96 + 1.07i)5-s + (3.64 − 3.64i)7-s + (−1.73 + 2.38i)9-s + (1.40 + 1.93i)11-s + (−3.87 + 0.613i)13-s + (0.482 + 0.0148i)15-s + (1.70 − 3.34i)17-s + (−1.33 − 4.11i)19-s + (0.343 − 1.05i)21-s + (4.03 + 0.639i)23-s + (2.68 + 4.21i)25-s + (−0.201 + 1.26i)27-s + (6.80 + 2.20i)29-s + (−4.89 + 1.59i)31-s + ⋯
L(s)  = 1  + (0.111 − 0.0565i)3-s + (0.876 + 0.481i)5-s + (1.37 − 1.37i)7-s + (−0.578 + 0.796i)9-s + (0.424 + 0.584i)11-s + (−1.07 + 0.170i)13-s + (0.124 + 0.00382i)15-s + (0.413 − 0.812i)17-s + (−0.306 − 0.943i)19-s + (0.0749 − 0.230i)21-s + (0.842 + 0.133i)23-s + (0.537 + 0.843i)25-s + (−0.0386 + 0.244i)27-s + (1.26 + 0.410i)29-s + (−0.879 + 0.285i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.999 + 0.0404i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.999 + 0.0404i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76542 - 0.0357271i\)
\(L(\frac12)\) \(\approx\) \(1.76542 - 0.0357271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.96 - 1.07i)T \)
good3 \( 1 + (-0.192 + 0.0980i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-3.64 + 3.64i)T - 7iT^{2} \)
11 \( 1 + (-1.40 - 1.93i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.87 - 0.613i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-1.70 + 3.34i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.33 + 4.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.03 - 0.639i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-6.80 - 2.20i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.89 - 1.59i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.32 - 8.37i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (3.51 + 2.55i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.236 - 0.236i)T + 43iT^{2} \)
47 \( 1 + (3.73 + 7.32i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.46 - 2.87i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (8.27 + 6.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.77 - 4.19i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.40 + 1.73i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (10.6 + 3.45i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.176 + 1.11i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (3.53 - 10.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.66 - 13.0i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (0.149 + 0.205i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.59 - 3.36i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.07605929083897788310651040782, −10.47880936960369514999199053862, −9.597884921804391461470084202274, −8.450900723747691900668617306524, −7.31572332940249625975335473421, −6.91481588483754685992520267272, −5.13176783823505810136447716677, −4.68412056371514546992770547972, −2.84387012566062474706913331821, −1.57626044950546183195696344334, 1.61526672174308991861516772514, 2.83791028675272381642618399387, 4.58191425766586483681082166542, 5.63339594956120915901380736530, 6.12589132310620507355942058832, 7.85601087157856631904191732009, 8.725335539832111508317823902450, 9.180287691619963691052610705208, 10.31815774720477901422506409385, 11.43899962232558073790252476338

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