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Properties

Label 2-10e3-1000.109-c1-0-119
Degree $2$
Conductor $1000$
Sign $0.921 - 0.387i$
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  + (1.34 + 0.439i)2-s + (0.0625 + 0.0755i)3-s + (1.61 + 1.18i)4-s + (2.10 + 0.743i)5-s + (0.0508 + 0.129i)6-s + (0.464 − 0.150i)7-s + (1.65 + 2.29i)8-s + (0.560 − 2.93i)9-s + (2.50 + 1.92i)10-s + (0.237 − 1.88i)11-s + (0.0116 + 0.195i)12-s + (0.985 − 5.16i)13-s + (0.690 + 0.00123i)14-s + (0.0757 + 0.205i)15-s + (1.20 + 3.81i)16-s + (−0.807 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.950 + 0.310i)2-s + (0.0361 + 0.0436i)3-s + (0.806 + 0.590i)4-s + (0.943 + 0.332i)5-s + (0.0207 + 0.0527i)6-s + (0.175 − 0.0570i)7-s + (0.583 + 0.812i)8-s + (0.186 − 0.979i)9-s + (0.793 + 0.608i)10-s + (0.0717 − 0.567i)11-s + (0.00335 + 0.0565i)12-s + (0.273 − 1.43i)13-s + (0.184 + 0.000330i)14-s + (0.0195 + 0.0531i)15-s + (0.302 + 0.953i)16-s + (−0.195 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.921 - 0.387i$
Analytic conductor: \(7.98504\)
Root analytic conductor: \(2.82578\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1000} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1000,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.49642 + 0.705630i\)
\(L(\frac12)\) \(\approx\) \(3.49642 + 0.705630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.439i)T \)
5 \( 1 + (-2.10 - 0.743i)T \)
good3 \( 1 + (-0.0625 - 0.0755i)T + (-0.562 + 2.94i)T^{2} \)
7 \( 1 + (-0.464 + 0.150i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.237 + 1.88i)T + (-10.6 - 2.73i)T^{2} \)
13 \( 1 + (-0.985 + 5.16i)T + (-12.0 - 4.78i)T^{2} \)
17 \( 1 + (0.807 + 3.14i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (-3.68 - 3.05i)T + (3.56 + 18.6i)T^{2} \)
23 \( 1 + (4.54 - 4.84i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (5.64 - 0.355i)T + (28.7 - 3.63i)T^{2} \)
31 \( 1 + (-2.67 + 0.686i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (5.77 + 3.17i)T + (19.8 + 31.2i)T^{2} \)
41 \( 1 + (8.44 - 7.93i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (4.06 - 2.95i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-3.01 - 7.62i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (1.79 - 2.83i)T + (-22.5 - 47.9i)T^{2} \)
59 \( 1 + (-0.228 + 0.107i)T + (37.6 - 45.4i)T^{2} \)
61 \( 1 + (2.92 - 3.11i)T + (-3.83 - 60.8i)T^{2} \)
67 \( 1 + (-0.642 + 10.2i)T + (-66.4 - 8.39i)T^{2} \)
71 \( 1 + (2.70 - 1.06i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (-1.99 - 0.939i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (-10.8 - 13.0i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (-10.0 + 12.1i)T + (-15.5 - 81.5i)T^{2} \)
89 \( 1 + (-5.72 + 12.1i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (0.488 - 0.0307i)T + (96.2 - 12.1i)T^{2} \)
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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

−10.04676484177953071114452350383, −9.318703805842834249482030110118, −8.116770255370538815934057995832, −7.38340038309913420660099168308, −6.30245719601228843720768672172, −5.80051786634631685126012350084, −5.00159714374230223283714033053, −3.53971394537187361121531890574, −3.05143402429179461862995320877, −1.51206240813572795473644694014, 1.77468636998950885820529470025, 2.15670598354214714201514477732, 3.78013088768507205827260361600, 4.78263905719872397346575025211, 5.32242480647156433667002812947, 6.47684900163886334286353513595, 7.01810160582940848043676957490, 8.295900671969595283924422684543, 9.262418143040766093977066534340, 10.15674157883469409435797650145

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