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Properties

Label 2-10e3-1000.109-c1-0-115
Degree $2$
Conductor $1000$
Sign $-0.544 + 0.838i$
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  + (0.803 − 1.16i)2-s + (0.573 + 0.693i)3-s + (−0.708 − 1.87i)4-s + (−1.74 + 1.39i)5-s + (1.26 − 0.110i)6-s + (1.39 − 0.454i)7-s + (−2.74 − 0.677i)8-s + (0.410 − 2.15i)9-s + (0.214 + 3.15i)10-s + (−0.0505 + 0.400i)11-s + (0.890 − 1.56i)12-s + (0.305 − 1.60i)13-s + (0.594 − 1.99i)14-s + (−1.97 − 0.414i)15-s + (−2.99 + 2.65i)16-s + (−0.459 − 1.78i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (0.331 + 0.400i)3-s + (−0.354 − 0.935i)4-s + (−0.782 + 0.622i)5-s + (0.517 − 0.0451i)6-s + (0.528 − 0.171i)7-s + (−0.970 − 0.239i)8-s + (0.136 − 0.716i)9-s + (0.0677 + 0.997i)10-s + (−0.0152 + 0.120i)11-s + (0.257 − 0.451i)12-s + (0.0846 − 0.443i)13-s + (0.158 − 0.532i)14-s + (−0.508 − 0.107i)15-s + (−0.748 + 0.662i)16-s + (−0.111 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $-0.544 + 0.838i$
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: \((2,\ 1000,\ (\ :1/2),\ -0.544 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841942 - 1.55043i\)
\(L(\frac12)\) \(\approx\) \(0.841942 - 1.55043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.803 + 1.16i)T \)
5 \( 1 + (1.74 - 1.39i)T \)
good3 \( 1 + (-0.573 - 0.693i)T + (-0.562 + 2.94i)T^{2} \)
7 \( 1 + (-1.39 + 0.454i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.0505 - 0.400i)T + (-10.6 - 2.73i)T^{2} \)
13 \( 1 + (-0.305 + 1.60i)T + (-12.0 - 4.78i)T^{2} \)
17 \( 1 + (0.459 + 1.78i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (6.64 + 5.49i)T + (3.56 + 18.6i)T^{2} \)
23 \( 1 + (-4.53 + 4.83i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (-4.45 + 0.280i)T + (28.7 - 3.63i)T^{2} \)
31 \( 1 + (-8.19 + 2.10i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (0.0562 + 0.0309i)T + (19.8 + 31.2i)T^{2} \)
41 \( 1 + (2.08 - 1.96i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (7.23 - 5.25i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.62 + 6.63i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (-5.98 + 9.43i)T + (-22.5 - 47.9i)T^{2} \)
59 \( 1 + (-7.61 + 3.58i)T + (37.6 - 45.4i)T^{2} \)
61 \( 1 + (7.36 - 7.84i)T + (-3.83 - 60.8i)T^{2} \)
67 \( 1 + (0.696 - 11.0i)T + (-66.4 - 8.39i)T^{2} \)
71 \( 1 + (7.50 - 2.97i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (13.4 + 6.31i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (-0.0651 - 0.0787i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (-2.41 + 2.91i)T + (-15.5 - 81.5i)T^{2} \)
89 \( 1 + (2.19 - 4.66i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (-10.3 + 0.652i)T + (96.2 - 12.1i)T^{2} \)
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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

−10.10777857949703947627762422970, −8.830070360903266914205091055106, −8.370779611407691888288962097781, −6.90153958211639803358387959166, −6.37467613801095599761445008724, −4.71862475980895281609586134709, −4.41681072816111878798966112149, −3.23417832057121847880178915388, −2.53845153232655345270898225768, −0.66104948525347246913466857588, 1.71626956145433394484677457112, 3.21214303088139387669929339042, 4.35266475940973618313852048810, 4.89621304568470844748523502387, 6.00318285428594002823746369684, 6.99321751438692124247668803468, 7.81427013283922775281629508992, 8.431220356340648240199177184211, 8.806383842646618597458005758866, 10.27208478066659589571288324848

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