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Properties

Label 2-20e2-80.69-c1-0-28
Degree $2$
Conductor $400$
Sign $-0.997 + 0.0708i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1 − i)3-s − 2i·4-s + 2·6-s − 2·7-s + (2 + 2i)8-s − i·9-s + (1 + i)11-s + (−2 + 2i)12-s + (−1 − i)13-s + (2 − 2i)14-s − 4·16-s + 2i·17-s + (1 + i)18-s + (−3 + 3i)19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.577 − 0.577i)3-s − i·4-s + 0.816·6-s − 0.755·7-s + (0.707 + 0.707i)8-s − 0.333i·9-s + (0.301 + 0.301i)11-s + (−0.577 + 0.577i)12-s + (−0.277 − 0.277i)13-s + (0.534 − 0.534i)14-s − 16-s + 0.485i·17-s + (0.235 + 0.235i)18-s + (−0.688 + 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.997 + 0.0708i$
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: \(1\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.997 + 0.0708i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (5 - 5i)T - 43iT^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 + (9 - 9i)T - 61iT^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−10.62715082683469826464073059420, −9.826661866431389346779887392965, −9.000279385484718572602242679521, −7.895921455341888060815499456993, −6.97054748817549145268800107900, −6.24154080623458419056170767990, −5.50419091175507436436231964851, −3.86385342093561225876920179955, −1.77798499173127444852298188332, 0, 2.21855833427292704335805704474, 3.63713117883414593857642957260, 4.66285355713816128156926393374, 6.04921895578873462321944675501, 7.16852640369284220072545235432, 8.239565037374621293852722793657, 9.365213889626572222592258006465, 9.868540195777612973233646985484, 10.85484159657115343683812389551

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