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Properties

Label 2-20e2-25.11-c1-0-3
Degree $2$
Conductor $400$
Sign $0.728 - 0.684i$
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  + (−0.309 + 0.951i)3-s + (−1.80 − 1.31i)5-s + 1.61·7-s + (1.61 + 1.17i)9-s + (−0.618 + 0.449i)11-s + (3.92 + 2.85i)13-s + (1.80 − 1.31i)15-s + (−0.236 − 0.726i)17-s + (1.80 + 5.56i)19-s + (−0.500 + 1.53i)21-s + (6.66 − 4.84i)23-s + (1.54 + 4.75i)25-s + (−4.04 + 2.93i)27-s + (−0.427 + 1.31i)29-s + (0.927 + 2.85i)31-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (−0.809 − 0.587i)5-s + 0.611·7-s + (0.539 + 0.391i)9-s + (−0.186 + 0.135i)11-s + (1.08 + 0.791i)13-s + (0.467 − 0.339i)15-s + (−0.0572 − 0.176i)17-s + (0.415 + 1.27i)19-s + (−0.109 + 0.335i)21-s + (1.38 − 1.00i)23-s + (0.309 + 0.951i)25-s + (−0.778 + 0.565i)27-s + (−0.0793 + 0.244i)29-s + (0.166 + 0.512i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.728 - 0.684i$
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: \((2,\ 400,\ (\ :1/2),\ 0.728 - 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19004 + 0.471172i\)
\(L(\frac12)\) \(\approx\) \(1.19004 + 0.471172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 + 1.31i)T \)
good3 \( 1 + (0.309 - 0.951i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + (0.618 - 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.92 - 2.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.236 + 0.726i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.80 - 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.66 + 4.84i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.427 - 1.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.927 - 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.42 + 2.48i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (-0.5 + 1.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.69 + 5.20i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.35 + 2.43i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.80 + 2.76i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.85 + 8.78i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.35 + 4.16i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.28 + 5.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.954 - 2.93i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.545 - 1.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.23 - 5.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.881 + 2.71i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.19860383210145673895186318838, −10.71957946353132194982336524511, −9.522438865143509798639682805006, −8.594469953083785914031829487276, −7.83159328398851332861723443170, −6.75318775289623251049041955517, −5.26225800036904088435028231082, −4.52791246929783635724231948826, −3.58430763894593814864241936193, −1.49123253415542256857255325216, 1.05608061582982353773447336490, 2.95783075270479551578787472915, 4.10055862822694626467607590274, 5.42367596477478542584421063494, 6.64572799923605973360535853442, 7.39927678304231182769218131104, 8.186559950057999961646734575604, 9.249978242049803758474081671068, 10.56779033729536545418583125699, 11.24097343623516080118711857573

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