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Properties

Label 2-10e3-25.19-c1-0-12
Degree $2$
Conductor $1000$
Sign $-0.119 + 0.992i$
Analytic cond. $7.98504$
Root an. cond. $2.82578$
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  + (−1.76 − 2.42i)3-s + 4.21i·7-s + (−1.84 + 5.68i)9-s + (−0.910 − 2.80i)11-s + (3.05 + 0.992i)13-s + (2.94 − 4.05i)17-s + (−3.38 − 2.45i)19-s + (10.2 − 7.42i)21-s + (2.02 − 0.658i)23-s + (8.48 − 2.75i)27-s + (5.74 − 4.17i)29-s + (0.918 + 0.667i)31-s + (−5.19 + 7.14i)33-s + (1.94 + 0.630i)37-s + (−2.97 − 9.15i)39-s + ⋯
L(s)  = 1  + (−1.01 − 1.39i)3-s + 1.59i·7-s + (−0.615 + 1.89i)9-s + (−0.274 − 0.845i)11-s + (0.847 + 0.275i)13-s + (0.715 − 0.984i)17-s + (−0.775 − 0.563i)19-s + (2.22 − 1.61i)21-s + (0.422 − 0.137i)23-s + (1.63 − 0.530i)27-s + (1.06 − 0.775i)29-s + (0.165 + 0.119i)31-s + (−0.903 + 1.24i)33-s + (0.319 + 0.103i)37-s + (−0.476 − 1.46i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.119 + 0.992i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.119 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653625 - 0.736727i\)
\(L(\frac12)\) \(\approx\) \(0.653625 - 0.736727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.76 + 2.42i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 - 4.21iT - 7T^{2} \)
11 \( 1 + (0.910 + 2.80i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.05 - 0.992i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.94 + 4.05i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.38 + 2.45i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.02 + 0.658i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.74 + 4.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.918 - 0.667i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.94 - 0.630i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.323 - 0.996i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.45iT - 43T^{2} \)
47 \( 1 + (3.94 + 5.42i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.86 + 2.56i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.87 + 8.85i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.556 - 1.71i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.79 + 6.59i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.16 + 5.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.00 - 0.327i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.21 - 4.51i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.11 - 8.42i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.68 - 5.18i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (9.97 + 13.7i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.727405503884059727081664388367, −8.557402779782587096305316553727, −8.221318321813945367338547262120, −6.97183778129332800511215601875, −6.28236511250522992479402766445, −5.66958875824618303275071744905, −4.92224216039856574654021676013, −3.00379003283801769505904423689, −2.03763451820847766597908120056, −0.62823669949900980924825693629, 1.12232292002471409347888857408, 3.40471948271598021398776664271, 4.15244847715101443368013386265, 4.75410473325003961959095547543, 5.83523077244001221684329259340, 6.60717740289816626607452249335, 7.66119467798209860787249853409, 8.658518767308475384137107126045, 9.863434139315543672783154506677, 10.29166170412923682958443324334

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