Properties

Label 1-2011-2011.12-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.115 - 0.993i$
Analytic cond. $216.111$
Root an. cond. $216.111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.578 − 0.815i)2-s + (0.0359 + 0.999i)3-s + (−0.329 − 0.944i)4-s + (0.631 + 0.775i)5-s + (0.835 + 0.549i)6-s + (−0.849 + 0.528i)7-s + (−0.960 − 0.277i)8-s + (−0.997 + 0.0718i)9-s + (0.997 − 0.0655i)10-s + (−0.408 + 0.912i)11-s + (0.931 − 0.363i)12-s + (−0.850 + 0.525i)13-s + (−0.0609 + 0.998i)14-s + (−0.752 + 0.658i)15-s + (−0.782 + 0.622i)16-s + (0.0920 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.578 − 0.815i)2-s + (0.0359 + 0.999i)3-s + (−0.329 − 0.944i)4-s + (0.631 + 0.775i)5-s + (0.835 + 0.549i)6-s + (−0.849 + 0.528i)7-s + (−0.960 − 0.277i)8-s + (−0.997 + 0.0718i)9-s + (0.997 − 0.0655i)10-s + (−0.408 + 0.912i)11-s + (0.931 − 0.363i)12-s + (−0.850 + 0.525i)13-s + (−0.0609 + 0.998i)14-s + (−0.752 + 0.658i)15-s + (−0.782 + 0.622i)16-s + (0.0920 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.115 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1565159502 - 0.1756832824i\)
\(L(\frac12)\) \(\approx\) \(0.1565159502 - 0.1756832824i\)
\(L(1)\) \(\approx\) \(1.005922454 + 0.1406533615i\)
\(L(1)\) \(\approx\) \(1.005922454 + 0.1406533615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.578 - 0.815i)T \)
3 \( 1 + (0.0359 + 0.999i)T \)
5 \( 1 + (0.631 + 0.775i)T \)
7 \( 1 + (-0.849 + 0.528i)T \)
11 \( 1 + (-0.408 + 0.912i)T \)
13 \( 1 + (-0.850 + 0.525i)T \)
17 \( 1 + (0.0920 - 0.995i)T \)
19 \( 1 + (0.823 + 0.567i)T \)
23 \( 1 + (-0.950 - 0.310i)T \)
29 \( 1 + (-0.338 - 0.940i)T \)
31 \( 1 + (0.621 + 0.783i)T \)
37 \( 1 + (-0.486 + 0.873i)T \)
41 \( 1 + (-0.987 - 0.158i)T \)
43 \( 1 + (-0.521 - 0.853i)T \)
47 \( 1 + (-0.464 + 0.885i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.984 + 0.177i)T \)
61 \( 1 + (-0.230 + 0.973i)T \)
67 \( 1 + (-0.101 + 0.994i)T \)
71 \( 1 + (0.934 - 0.354i)T \)
73 \( 1 + (-0.957 + 0.289i)T \)
79 \( 1 + (-0.818 + 0.575i)T \)
83 \( 1 + (0.350 - 0.936i)T \)
89 \( 1 + (-0.662 - 0.749i)T \)
97 \( 1 + (-0.792 - 0.610i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.94295069888793975883998277074, −19.28901905057095570112317467814, −18.218333577480745994754101563761, −17.65433594337558410149327376938, −16.92443287836099520457991634398, −16.47402819714715335030667025013, −15.68447089538743454987232317640, −14.64914781670607076343940706516, −13.84396671017832842174222701495, −13.39087866063000354675365789076, −12.83038970228299361482534012841, −12.30051999913052344538943365973, −11.37141297745248224104057641270, −10.11602916310573955704097565065, −9.32363317585006119777248847191, −8.36463193552687960085476953821, −7.9057789124962138288185803603, −6.97944693501981566440925478499, −6.281642330636615083352758770300, −5.59659654301007913489505996547, −5.01646463827371601549314172595, −3.6819775366489128512475536371, −3.00445389122503899867292377840, −1.9606828088746774626039794365, −0.65917819378169779178224200197, 0.04354466720911265851591857812, 1.81874869748324928914340156681, 2.67048980859419579211028528117, 3.06356362775994891460946378949, 4.08629995086881951229045813798, 4.96764630701083394260532284648, 5.58587590564361342454402577625, 6.40170404931146066426605239199, 7.29342132720835846936822529084, 8.75143320365801887090937114038, 9.63775291961197235691441967403, 10.01235037735455085484688768163, 10.28932082241715055383635150378, 11.73870470876084155641065166609, 11.82216772729531972872831993833, 12.92877101268046172632949986314, 13.96729073724321813239390070529, 14.18828571064725146761541733034, 15.23623530180380839871769403692, 15.55703232742635483669138926370, 16.5211477385238256712934540310, 17.52323033056647741385596552717, 18.29450320491414576827875245811, 18.9336045121339587531978214869, 19.717519753470471748296897070889

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