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Properties

Label 2-20e2-100.23-c1-0-3
Degree $2$
Conductor $400$
Sign $0.916 - 0.400i$
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  + (−0.157 − 0.308i)3-s + (−2.16 + 0.548i)5-s + (1.64 + 1.64i)7-s + (1.69 − 2.32i)9-s + (2.02 + 2.78i)11-s + (−0.333 − 2.10i)13-s + (0.510 + 0.583i)15-s + (4.32 + 2.20i)17-s + (2.39 + 7.35i)19-s + (0.249 − 0.768i)21-s + (−0.0697 + 0.440i)23-s + (4.39 − 2.37i)25-s + (−2.01 − 0.318i)27-s + (1.71 + 0.557i)29-s + (7.04 − 2.29i)31-s + ⋯
L(s)  = 1  + (−0.0908 − 0.178i)3-s + (−0.969 + 0.245i)5-s + (0.622 + 0.622i)7-s + (0.564 − 0.776i)9-s + (0.610 + 0.840i)11-s + (−0.0924 − 0.583i)13-s + (0.131 + 0.150i)15-s + (1.04 + 0.534i)17-s + (0.548 + 1.68i)19-s + (0.0544 − 0.167i)21-s + (−0.0145 + 0.0918i)23-s + (0.879 − 0.475i)25-s + (−0.387 − 0.0613i)27-s + (0.318 + 0.103i)29-s + (1.26 − 0.411i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.916 - 0.400i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27351 + 0.265845i\)
\(L(\frac12)\) \(\approx\) \(1.27351 + 0.265845i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 0.548i)T \)
good3 \( 1 + (0.157 + 0.308i)T + (-1.76 + 2.42i)T^{2} \)
7 \( 1 + (-1.64 - 1.64i)T + 7iT^{2} \)
11 \( 1 + (-2.02 - 2.78i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.333 + 2.10i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.32 - 2.20i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.39 - 7.35i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.0697 - 0.440i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-1.71 - 0.557i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-7.04 + 2.29i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.39 - 0.696i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (2.90 + 2.11i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-7.50 + 7.50i)T - 43iT^{2} \)
47 \( 1 + (11.3 - 5.75i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (7.81 - 3.98i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (1.55 + 1.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.52 + 4.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.31 - 4.54i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-3.94 - 1.28i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.07 + 0.804i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.36 - 4.20i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.6 + 5.93i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-8.50 - 11.7i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.61 - 9.05i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.63252055981463431057946757193, −10.36088463110777312109201963067, −9.640324132995886333485040888380, −8.323082129857243171584418654237, −7.72899754359329060999713330791, −6.70971876741815432379492634078, −5.58601485664488926980258843503, −4.28999943096802172444730611394, −3.32905734555450660609027554983, −1.45354421136229280070333401152, 1.08945693697388619609995900829, 3.17296112133096262299108421610, 4.44593055145446764188422257802, 5.01688194508339006488631428547, 6.70921468397675721570863392841, 7.55906157012189275455610218067, 8.323810235442913591028068190880, 9.368579611113067307509155317660, 10.46122603403067907306765209108, 11.41617508607063343561580355999

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