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Properties

Label 2-40e2-80.27-c1-0-24
Degree $2$
Conductor $1600$
Sign $0.999 - 0.0400i$
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  + 3.07·3-s + (1.47 + 1.47i)7-s + 6.45·9-s + (1.20 − 1.20i)11-s − 5.63i·13-s + (4.22 + 4.22i)17-s + (−3.11 + 3.11i)19-s + (4.54 + 4.54i)21-s + (−1.08 + 1.08i)23-s + 10.6·27-s + (−5.32 − 5.32i)29-s + 4.67i·31-s + (3.71 − 3.71i)33-s − 1.51i·37-s − 17.3i·39-s + ⋯
L(s)  = 1  + 1.77·3-s + (0.558 + 0.558i)7-s + 2.15·9-s + (0.363 − 0.363i)11-s − 1.56i·13-s + (1.02 + 1.02i)17-s + (−0.715 + 0.715i)19-s + (0.991 + 0.991i)21-s + (−0.225 + 0.225i)23-s + 2.04·27-s + (−0.988 − 0.988i)29-s + 0.839i·31-s + (0.646 − 0.646i)33-s − 0.248i·37-s − 2.77i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.999 - 0.0400i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.999 - 0.0400i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.557554008\)
\(L(\frac12)\) \(\approx\) \(3.557554008\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.07T + 3T^{2} \)
7 \( 1 + (-1.47 - 1.47i)T + 7iT^{2} \)
11 \( 1 + (-1.20 + 1.20i)T - 11iT^{2} \)
13 \( 1 + 5.63iT - 13T^{2} \)
17 \( 1 + (-4.22 - 4.22i)T + 17iT^{2} \)
19 \( 1 + (3.11 - 3.11i)T - 19iT^{2} \)
23 \( 1 + (1.08 - 1.08i)T - 23iT^{2} \)
29 \( 1 + (5.32 + 5.32i)T + 29iT^{2} \)
31 \( 1 - 4.67iT - 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 + 3.19iT - 41T^{2} \)
43 \( 1 + 2.42iT - 43T^{2} \)
47 \( 1 + (0.827 - 0.827i)T - 47iT^{2} \)
53 \( 1 + 8.17T + 53T^{2} \)
59 \( 1 + (-7.78 - 7.78i)T + 59iT^{2} \)
61 \( 1 + (3.03 - 3.03i)T - 61iT^{2} \)
67 \( 1 - 2.93iT - 67T^{2} \)
71 \( 1 - 0.180T + 71T^{2} \)
73 \( 1 + (2.19 + 2.19i)T + 73iT^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + 8.33T + 83T^{2} \)
89 \( 1 - 9.08T + 89T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−9.194893611938556231341861625535, −8.462844909588221856882642505676, −8.037560943210935270771180100679, −7.48393586833972277831547071743, −6.12447570165414064860163992239, −5.31962699478469174226972904952, −3.94783405640365848019346141181, −3.41718304585976550564715731008, −2.40166820721442386536169150649, −1.45739631010350081514984668945, 1.46715554648824207600370593963, 2.29872520376002400082366996531, 3.38902396858259835291833177574, 4.22179230025744346892864929053, 4.87149850434408750537857765102, 6.54534254263020026703170508281, 7.25260481740936516357846242872, 7.83326423911230229384616057307, 8.652288994655516374052104732339, 9.449656274908462258445861215895

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