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Properties

Label 2-10e3-25.4-c1-0-21
Degree $2$
Conductor $1000$
Sign $-0.628 + 0.777i$
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  + (1.98 − 2.72i)3-s − 0.794i·7-s + (−2.58 − 7.94i)9-s + (0.673 − 2.07i)11-s + (2.73 − 0.888i)13-s + (−0.0736 − 0.101i)17-s + (−5.66 + 4.11i)19-s + (−2.16 − 1.57i)21-s + (5.38 + 1.74i)23-s + (−17.1 − 5.57i)27-s + (0.989 + 0.718i)29-s + (−5.53 + 4.01i)31-s + (−4.31 − 5.94i)33-s + (6.55 − 2.12i)37-s + (2.99 − 9.21i)39-s + ⋯
L(s)  = 1  + (1.14 − 1.57i)3-s − 0.300i·7-s + (−0.860 − 2.64i)9-s + (0.203 − 0.624i)11-s + (0.758 − 0.246i)13-s + (−0.0178 − 0.0245i)17-s + (−1.29 + 0.943i)19-s + (−0.472 − 0.343i)21-s + (1.12 + 0.364i)23-s + (−3.30 − 1.07i)27-s + (0.183 + 0.133i)29-s + (−0.993 + 0.721i)31-s + (−0.751 − 1.03i)33-s + (1.07 − 0.350i)37-s + (0.479 − 1.47i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.628 + 0.777i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ -0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970304 - 2.03260i\)
\(L(\frac12)\) \(\approx\) \(0.970304 - 2.03260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 + 2.72i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.794iT - 7T^{2} \)
11 \( 1 + (-0.673 + 2.07i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.73 + 0.888i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.0736 + 0.101i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.66 - 4.11i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.38 - 1.74i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.989 - 0.718i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.53 - 4.01i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.55 + 2.12i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.244 - 0.753i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.69iT - 43T^{2} \)
47 \( 1 + (1.79 - 2.46i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.536 + 0.738i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.853 - 2.62i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.22 + 9.92i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.90 - 3.99i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.39 + 4.64i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.01 + 0.980i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.78 - 1.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.280 - 0.385i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.61 - 14.2i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.06 + 6.97i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.262482663483781438846877059381, −8.662606087830128379651997720355, −8.060223393021155012837582519229, −7.22657489364056970653449198944, −6.49525904130617708140136815217, −5.71964732655714333096403700504, −3.90477571876458279540926014913, −3.16056885635640287419348540109, −1.98607454804731189775819142244, −0.932657599038048472474150248139, 2.16373691864068546523066903074, 3.05325462393104379251824679035, 4.18298905347806903505279946108, 4.62649369372463264886361141964, 5.77652475044411611596766203036, 7.03943212827518439710998206203, 8.151910946723141702748591509564, 8.858287592140459419388498602940, 9.287440349047988971501157610534, 10.13693437938864532862904731015

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