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Properties

Label 1-2011-2011.26-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.927 - 0.373i$
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  + (0.854 + 0.520i)2-s + (0.857 + 0.514i)3-s + (0.458 + 0.888i)4-s + (0.659 + 0.751i)5-s + (0.464 + 0.885i)6-s + (−0.984 − 0.174i)7-s + (−0.0702 + 0.997i)8-s + (0.469 + 0.882i)9-s + (0.172 + 0.984i)10-s + (−0.0265 + 0.999i)11-s + (−0.0640 + 0.997i)12-s + (0.326 − 0.945i)13-s + (−0.750 − 0.660i)14-s + (0.178 + 0.983i)15-s + (−0.578 + 0.815i)16-s + (0.666 − 0.745i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.520i)2-s + (0.857 + 0.514i)3-s + (0.458 + 0.888i)4-s + (0.659 + 0.751i)5-s + (0.464 + 0.885i)6-s + (−0.984 − 0.174i)7-s + (−0.0702 + 0.997i)8-s + (0.469 + 0.882i)9-s + (0.172 + 0.984i)10-s + (−0.0265 + 0.999i)11-s + (−0.0640 + 0.997i)12-s + (0.326 − 0.945i)13-s + (−0.750 − 0.660i)14-s + (0.178 + 0.983i)15-s + (−0.578 + 0.815i)16-s + (0.666 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.019453281 + 5.268290102i\)
\(L(\frac12)\) \(\approx\) \(-1.019453281 + 5.268290102i\)
\(L(1)\) \(\approx\) \(1.521274624 + 1.810597190i\)
\(L(1)\) \(\approx\) \(1.521274624 + 1.810597190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.854 + 0.520i)T \)
3 \( 1 + (0.857 + 0.514i)T \)
5 \( 1 + (0.659 + 0.751i)T \)
7 \( 1 + (-0.984 - 0.174i)T \)
11 \( 1 + (-0.0265 + 0.999i)T \)
13 \( 1 + (0.326 - 0.945i)T \)
17 \( 1 + (0.666 - 0.745i)T \)
19 \( 1 + (0.812 + 0.582i)T \)
23 \( 1 + (-0.233 + 0.972i)T \)
29 \( 1 + (-0.581 + 0.813i)T \)
31 \( 1 + (0.0889 + 0.996i)T \)
37 \( 1 + (0.964 - 0.262i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (-0.976 + 0.214i)T \)
47 \( 1 + (0.0421 - 0.999i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.413 - 0.910i)T \)
61 \( 1 + (-0.0203 - 0.999i)T \)
67 \( 1 + (0.359 + 0.933i)T \)
71 \( 1 + (0.858 + 0.512i)T \)
73 \( 1 + (-0.645 - 0.763i)T \)
79 \( 1 + (-0.889 - 0.457i)T \)
83 \( 1 + (-0.0109 + 0.999i)T \)
89 \( 1 + (0.932 + 0.360i)T \)
97 \( 1 + (-0.163 + 0.986i)T \)
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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

−19.535052467636783642535642538522, −18.64368547376259498669333855394, −18.51960260929540501565979596158, −16.91023166091426430170552913881, −16.38254943498256517141903161847, −15.555007119295846557741127937774, −14.72394749066035798710493876702, −13.87068212176793342865544891552, −13.43306040397930133110399885883, −12.99568738571297516424707309697, −12.14856455859177536125942497925, −11.5466399720307255979662885667, −10.294655582096435484977114402479, −9.608207585914349981442170366273, −9.05001905940876235688289262505, −8.21372324775497515999023413568, −7.04321747127041844014037217076, −6.0464313908804427917153395937, −5.948856657314249782626768230991, −4.541923711820052511538548181764, −3.77187720376643906914024587119, −2.962368322211124278067767336467, −2.23795113876513897866988115490, −1.29921639989474335589856707738, −0.52994999706483930180459631712, 1.62542229508432194767839822834, 2.607387031393408796100156686407, 3.42963610052819389938758735027, 3.589010856120566044521030973041, 5.08383646614159221827182961543, 5.499359345146659102120158808353, 6.63294830182092565438506690235, 7.28538824564044178359842718330, 7.86042268972611291874827659446, 9.018413790200003322804595560922, 9.9197399527218208035849739237, 10.19237450675773628985891661025, 11.33082790715053709959155399536, 12.38331923834409178642018950959, 13.1371124910149360708810262168, 13.65414235630883743076886627505, 14.36053849140606928824516494523, 14.97229466710264199569099915759, 15.64359506050730848956587983792, 16.2331787409349444989568881865, 17.03125469337296209737629798047, 18.03005828790373748962616256663, 18.58452352441032503251155619569, 19.84108431868491320052680544743, 20.24584696138471558426811748587

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