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Properties

Label 2-40e2-80.27-c1-0-10
Degree $2$
Conductor $1600$
Sign $-0.0692 - 0.997i$
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  + 0.614·3-s + (2.83 + 2.83i)7-s − 2.62·9-s + (−1.95 + 1.95i)11-s − 2.05i·13-s + (4.06 + 4.06i)17-s + (0.683 − 0.683i)19-s + (1.74 + 1.74i)21-s + (−4.95 + 4.95i)23-s − 3.45·27-s + (0.835 + 0.835i)29-s − 2.35i·31-s + (−1.20 + 1.20i)33-s + 4.54i·37-s − 1.26i·39-s + ⋯
L(s)  = 1  + 0.354·3-s + (1.07 + 1.07i)7-s − 0.874·9-s + (−0.590 + 0.590i)11-s − 0.569i·13-s + (0.986 + 0.986i)17-s + (0.156 − 0.156i)19-s + (0.380 + 0.380i)21-s + (−1.03 + 1.03i)23-s − 0.664·27-s + (0.155 + 0.155i)29-s − 0.423i·31-s + (−0.209 + 0.209i)33-s + 0.747i·37-s − 0.202i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0692 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.0692 - 0.997i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.0692 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.652588698\)
\(L(\frac12)\) \(\approx\) \(1.652588698\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.614T + 3T^{2} \)
7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (-0.683 + 0.683i)T - 19iT^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + (-0.835 - 0.835i)T + 29iT^{2} \)
31 \( 1 + 2.35iT - 31T^{2} \)
37 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 - 0.849iT - 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 + 5.17T + 53T^{2} \)
59 \( 1 + (-4.16 - 4.16i)T + 59iT^{2} \)
61 \( 1 + (-5.55 + 5.55i)T - 61iT^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 2.75T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-3.52 - 3.52i)T + 97iT^{2} \)
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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

−9.603316679020344603319993581945, −8.605359923484143632029830247000, −8.061527639339067084691127430136, −7.63284873664286742910469866350, −6.11024134927337344930420127560, −5.53585283917434013272337928777, −4.81517102733054164911997117982, −3.50715277412499077867520157988, −2.56030415105728610554441610325, −1.62457403498381607244629049666, 0.59886030041197770889354750574, 2.04167084066531176799119340790, 3.14979251458643573636994803212, 4.11349992039554827060211052780, 5.05126464737572951495337104862, 5.82200556544381982401902355469, 6.97789674955243984984758306075, 7.78479300965852139616312223085, 8.265006976110090696126403463766, 9.063659166911011459711068636143

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