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Properties

Label 2-40e2-80.43-c1-0-8
Degree $2$
Conductor $1600$
Sign $-0.788 - 0.614i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + 1.28i·3-s + (−1.13 + 1.13i)7-s + 1.35·9-s + (2.32 − 2.32i)11-s − 1.36·13-s + (−5.25 + 5.25i)17-s + (−3.69 + 3.69i)19-s + (−1.46 − 1.46i)21-s + (−0.911 − 0.911i)23-s + 5.58i·27-s + (2.37 + 2.37i)29-s − 0.242i·31-s + (2.97 + 2.97i)33-s + 3.34·37-s − 1.74i·39-s + ⋯
L(s)  = 1  + 0.739i·3-s + (−0.430 + 0.430i)7-s + 0.452·9-s + (0.700 − 0.700i)11-s − 0.378·13-s + (−1.27 + 1.27i)17-s + (−0.848 + 0.848i)19-s + (−0.318 − 0.318i)21-s + (−0.189 − 0.189i)23-s + 1.07i·27-s + (0.440 + 0.440i)29-s − 0.0435i·31-s + (0.517 + 0.517i)33-s + 0.549·37-s − 0.280i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.788 - 0.614i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.788 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.133916373\)
\(L(\frac12)\) \(\approx\) \(1.133916373\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 1.28iT - 3T^{2} \)
7 \( 1 + (1.13 - 1.13i)T - 7iT^{2} \)
11 \( 1 + (-2.32 + 2.32i)T - 11iT^{2} \)
13 \( 1 + 1.36T + 13T^{2} \)
17 \( 1 + (5.25 - 5.25i)T - 17iT^{2} \)
19 \( 1 + (3.69 - 3.69i)T - 19iT^{2} \)
23 \( 1 + (0.911 + 0.911i)T + 23iT^{2} \)
29 \( 1 + (-2.37 - 2.37i)T + 29iT^{2} \)
31 \( 1 + 0.242iT - 31T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 - 9.04T + 43T^{2} \)
47 \( 1 + (7.87 + 7.87i)T + 47iT^{2} \)
53 \( 1 - 5.80iT - 53T^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 + 4.54T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 3.26iT - 83T^{2} \)
89 \( 1 + 9.77T + 89T^{2} \)
97 \( 1 + (-1.63 + 1.63i)T - 97iT^{2} \)
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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

−9.749522658514160224040542972719, −8.873750144459003895769557686206, −8.478053306538235966838294797863, −7.25630618917355956422783371022, −6.31369540531349261754781554670, −5.80113035221615224836650557866, −4.41542560258260123424611000332, −4.05387129044561438304609089446, −2.89517571764111914515003359915, −1.60580629460260324602259683722, 0.42937879650806101997645413729, 1.83866505292615319384160575233, 2.76921085889070684321181003542, 4.24270623140271026829477368390, 4.67998425517585041192661538540, 6.14677580098471315051710573740, 6.93914874143427254398896042396, 7.14925865599923936349634084999, 8.196695804253201942011853816388, 9.261629378646558641434878466938

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