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Properties

Label 2-10e3-1000.109-c1-0-112
Degree $2$
Conductor $1000$
Sign $0.922 + 0.386i$
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  + (1.10 − 0.888i)2-s + (1.24 + 1.50i)3-s + (0.422 − 1.95i)4-s + (−0.0625 + 2.23i)5-s + (2.70 + 0.549i)6-s + (2.70 − 0.878i)7-s + (−1.27 − 2.52i)8-s + (−0.149 + 0.784i)9-s + (1.91 + 2.51i)10-s + (0.494 − 3.91i)11-s + (3.46 − 1.79i)12-s + (0.919 − 4.82i)13-s + (2.19 − 3.36i)14-s + (−3.43 + 2.68i)15-s + (−3.64 − 1.65i)16-s + (0.499 + 1.94i)17-s + ⋯
L(s)  = 1  + (0.778 − 0.628i)2-s + (0.717 + 0.867i)3-s + (0.211 − 0.977i)4-s + (−0.0279 + 0.999i)5-s + (1.10 + 0.224i)6-s + (1.02 − 0.332i)7-s + (−0.449 − 0.893i)8-s + (−0.0499 + 0.261i)9-s + (0.605 + 0.795i)10-s + (0.149 − 1.17i)11-s + (0.999 − 0.517i)12-s + (0.255 − 1.33i)13-s + (0.586 − 0.900i)14-s + (−0.886 + 0.692i)15-s + (−0.910 − 0.412i)16-s + (0.121 + 0.472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1000\)    =    \(2^{3} \cdot 5^{3}\)
Sign: $0.922 + 0.386i$
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: \((2,\ 1000,\ (\ :1/2),\ 0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.37755 - 0.679299i\)
\(L(\frac12)\) \(\approx\) \(3.37755 - 0.679299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 + 0.888i)T \)
5 \( 1 + (0.0625 - 2.23i)T \)
good3 \( 1 + (-1.24 - 1.50i)T + (-0.562 + 2.94i)T^{2} \)
7 \( 1 + (-2.70 + 0.878i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.494 + 3.91i)T + (-10.6 - 2.73i)T^{2} \)
13 \( 1 + (-0.919 + 4.82i)T + (-12.0 - 4.78i)T^{2} \)
17 \( 1 + (-0.499 - 1.94i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (-4.73 - 3.91i)T + (3.56 + 18.6i)T^{2} \)
23 \( 1 + (4.66 - 4.96i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (-2.75 + 0.173i)T + (28.7 - 3.63i)T^{2} \)
31 \( 1 + (7.19 - 1.84i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (-5.26 - 2.89i)T + (19.8 + 31.2i)T^{2} \)
41 \( 1 + (0.720 - 0.676i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (4.55 - 3.31i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-3.72 - 9.41i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (1.03 - 1.63i)T + (-22.5 - 47.9i)T^{2} \)
59 \( 1 + (-6.34 + 2.98i)T + (37.6 - 45.4i)T^{2} \)
61 \( 1 + (3.27 - 3.49i)T + (-3.83 - 60.8i)T^{2} \)
67 \( 1 + (-0.775 + 12.3i)T + (-66.4 - 8.39i)T^{2} \)
71 \( 1 + (10.9 - 4.33i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (6.14 + 2.89i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (10.6 + 12.8i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (-2.98 + 3.60i)T + (-15.5 - 81.5i)T^{2} \)
89 \( 1 + (0.929 - 1.97i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (2.74 - 0.172i)T + (96.2 - 12.1i)T^{2} \)
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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

−10.18091071227267523754313958036, −9.415917331939092937766980050162, −8.218962267232715133091951313273, −7.59561048799199612636666397779, −6.11356039453423043503080777380, −5.54308442058279035969976774767, −4.29163301931955485092247035568, −3.39160171101150717260900993313, −3.07010348070351065303204036867, −1.42266096783668394116389254366, 1.69873303920350591282792159884, 2.41163106112957579407370669154, 4.11786654911652475567509386006, 4.75721774608362439764073468180, 5.57492640665259279592544825387, 6.97260463978261664505879853029, 7.32978897582589727299805096649, 8.327234516337968659848834364346, 8.748306110502611293485085831014, 9.653426458019372126766021089681

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