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Properties

Label 2-20e2-80.69-c1-0-14
Degree $2$
Conductor $400$
Sign $-0.150 - 0.988i$
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  + (1.21 + 0.719i)2-s + (1.66 + 1.66i)3-s + (0.965 + 1.75i)4-s + (0.831 + 3.23i)6-s − 1.87·7-s + (−0.0835 + 2.82i)8-s + 2.56i·9-s + (−3.29 − 3.29i)11-s + (−1.31 + 4.53i)12-s + (1.90 + 1.90i)13-s + (−2.28 − 1.34i)14-s + (−2.13 + 3.38i)16-s − 2.57i·17-s + (−1.84 + 3.12i)18-s + (5.76 − 5.76i)19-s + ⋯
L(s)  = 1  + (0.861 + 0.508i)2-s + (0.962 + 0.962i)3-s + (0.482 + 0.875i)4-s + (0.339 + 1.31i)6-s − 0.708·7-s + (−0.0295 + 0.999i)8-s + 0.854i·9-s + (−0.994 − 0.994i)11-s + (−0.378 + 1.30i)12-s + (0.527 + 0.527i)13-s + (−0.609 − 0.360i)14-s + (−0.533 + 0.845i)16-s − 0.623i·17-s + (−0.434 + 0.735i)18-s + (1.32 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.150 - 0.988i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.150 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77333 + 2.06449i\)
\(L(\frac12)\) \(\approx\) \(1.77333 + 2.06449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.719i)T \)
5 \( 1 \)
good3 \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \)
7 \( 1 + 1.87T + 7T^{2} \)
11 \( 1 + (3.29 + 3.29i)T + 11iT^{2} \)
13 \( 1 + (-1.90 - 1.90i)T + 13iT^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + (-5.76 + 5.76i)T - 19iT^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + (6.45 - 6.45i)T - 29iT^{2} \)
31 \( 1 + 0.799T + 31T^{2} \)
37 \( 1 + (-2.69 + 2.69i)T - 37iT^{2} \)
41 \( 1 - 0.946iT - 41T^{2} \)
43 \( 1 + (0.829 - 0.829i)T - 43iT^{2} \)
47 \( 1 + 1.52iT - 47T^{2} \)
53 \( 1 + (6.97 - 6.97i)T - 53iT^{2} \)
59 \( 1 + (6.84 + 6.84i)T + 59iT^{2} \)
61 \( 1 + (6.87 - 6.87i)T - 61iT^{2} \)
67 \( 1 + (-3.73 - 3.73i)T + 67iT^{2} \)
71 \( 1 + 9.34iT - 71T^{2} \)
73 \( 1 - 0.886T + 73T^{2} \)
79 \( 1 + 3.07T + 79T^{2} \)
83 \( 1 + (0.989 + 0.989i)T + 83iT^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 - 7.16iT - 97T^{2} \)
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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

−11.37192272546572657084416188231, −10.80025236950282843422306823845, −9.297259195262853068611717977246, −8.977791221979654185445081663259, −7.75292838267526241305865853020, −6.81871372751816473621025747263, −5.51761670711562301150159185575, −4.65097789774633245662336697274, −3.20925757234621619697580545711, −3.02928996337832618596551792249, 1.54490548865466786584941805278, 2.76410633386467547448512926900, 3.59578450158288657466615727360, 5.16083754985205523470444598022, 6.22654689912367878187934221490, 7.33268514756196175004233737661, 7.971782107589423315984448970186, 9.397404916333446835703918951437, 10.12265234658587194464107319680, 11.16464370355584945079147334402

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