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Properties

Label 2-10e3-25.4-c1-0-1
Degree $2$
Conductor $1000$
Sign $-0.781 - 0.623i$
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  + (−0.565 + 0.777i)3-s − 0.364i·7-s + (0.641 + 1.97i)9-s + (−0.768 + 2.36i)11-s + (−1.74 + 0.565i)13-s + (−1.11 − 1.53i)17-s + (0.483 − 0.350i)19-s + (0.283 + 0.205i)21-s + (−2.93 − 0.953i)23-s + (−4.64 − 1.50i)27-s + (6.58 + 4.78i)29-s + (−8.25 + 6.00i)31-s + (−1.40 − 1.93i)33-s + (0.387 − 0.125i)37-s + (0.543 − 1.67i)39-s + ⋯
L(s)  = 1  + (−0.326 + 0.449i)3-s − 0.137i·7-s + (0.213 + 0.658i)9-s + (−0.231 + 0.712i)11-s + (−0.482 + 0.156i)13-s + (−0.270 − 0.372i)17-s + (0.110 − 0.0805i)19-s + (0.0618 + 0.0449i)21-s + (−0.611 − 0.198i)23-s + (−0.893 − 0.290i)27-s + (1.22 + 0.887i)29-s + (−1.48 + 1.07i)31-s + (−0.244 − 0.336i)33-s + (0.0636 − 0.0206i)37-s + (0.0870 − 0.267i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.781 - 0.623i$
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: \((2,\ 1000,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273089 + 0.780474i\)
\(L(\frac12)\) \(\approx\) \(0.273089 + 0.780474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.565 - 0.777i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 0.364iT - 7T^{2} \)
11 \( 1 + (0.768 - 2.36i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.74 - 0.565i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.11 + 1.53i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.483 + 0.350i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.93 + 0.953i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-6.58 - 4.78i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (8.25 - 6.00i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.387 + 0.125i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.00 + 3.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.94iT - 43T^{2} \)
47 \( 1 + (6.57 - 9.05i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.77 - 6.57i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.32 - 4.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.72 - 8.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (6.44 + 8.87i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.5 + 7.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (11.1 + 3.62i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (6.55 + 4.76i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.68 - 13.3i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.19 - 3.68i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.04 + 6.93i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.45356726441880312286517150780, −9.570866026366206940245967029190, −8.768898422768978004968006408876, −7.62106805260166787850126437050, −7.09627490636938716026441884969, −5.91709030971408065694449552488, −4.88285408904139276862282559792, −4.42338398836340772141106117780, −3.00318418446626273017467147886, −1.75606075221317576832235126126, 0.38127992194488984897913781140, 1.92112466187808199972544295638, 3.25507938133231167993440131168, 4.28650524535495543975713357183, 5.54592410060878694284651897753, 6.18041277231514394010895258742, 7.07351075702418093856406303915, 7.944203106974918561166680081012, 8.772260572743054287182935258713, 9.716889290422613928406339521684

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