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Properties

Label 2-20e2-25.11-c1-0-0
Degree $2$
Conductor $400$
Sign $-0.753 + 0.657i$
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  + (−1.02 + 3.15i)3-s + (−1.34 + 1.78i)5-s − 1.69·7-s + (−6.44 − 4.68i)9-s + (0.568 − 0.413i)11-s + (3.22 + 2.34i)13-s + (−4.24 − 6.07i)15-s + (0.538 + 1.65i)17-s + (−1.16 − 3.59i)19-s + (1.73 − 5.35i)21-s + (−0.281 + 0.204i)23-s + (−1.37 − 4.80i)25-s + (13.3 − 9.68i)27-s + (−1.77 + 5.44i)29-s + (1.28 + 3.96i)31-s + ⋯
L(s)  = 1  + (−0.590 + 1.81i)3-s + (−0.602 + 0.798i)5-s − 0.642·7-s + (−2.14 − 1.56i)9-s + (0.171 − 0.124i)11-s + (0.895 + 0.650i)13-s + (−1.09 − 1.56i)15-s + (0.130 + 0.401i)17-s + (−0.267 − 0.823i)19-s + (0.379 − 1.16i)21-s + (−0.0585 + 0.0425i)23-s + (−0.274 − 0.961i)25-s + (2.56 − 1.86i)27-s + (−0.328 + 1.01i)29-s + (0.231 + 0.712i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178835 - 0.477000i\)
\(L(\frac12)\) \(\approx\) \(0.178835 - 0.477000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.78i)T \)
good3 \( 1 + (1.02 - 3.15i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 1.69T + 7T^{2} \)
11 \( 1 + (-0.568 + 0.413i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.22 - 2.34i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.538 - 1.65i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.16 + 3.59i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.281 - 0.204i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.77 - 5.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.28 - 3.96i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.51 + 2.55i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.22 + 5.24i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.63T + 43T^{2} \)
47 \( 1 + (1.87 - 5.76i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.21 - 9.89i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.40 - 1.74i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.92 - 2.12i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.26 - 6.98i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.795 - 2.44i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.20 + 5.96i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.84 + 8.75i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.73 - 8.42i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.607 - 0.441i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.81 + 5.58i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.44560075749231203082084279081, −10.82980336217158533655250461062, −10.24908538717522627302536300795, −9.233770787515780077759968217192, −8.545727334607363775969367562416, −6.84088871855291725968083385322, −6.09151986174298040629926949349, −4.86546062292540384472793089094, −3.81034907209779853762147738781, −3.20051692202431792934348186150, 0.35832441740836404894347443187, 1.70657374318819464551601521890, 3.42071658550057438273266403814, 5.14524124027689840324426069482, 6.11488273065159699410757680197, 6.86227836286398274938708886918, 8.055505241546161509631975241158, 8.312560109400926216772197669672, 9.795848771868859494164148276026, 11.16013837678800959194750351863

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