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Properties

Label 1-2011-2011.102-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.783 + 0.621i$
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  + (−0.194 + 0.980i)2-s + (−0.279 + 0.960i)3-s + (−0.924 − 0.380i)4-s + (0.788 + 0.615i)5-s + (−0.887 − 0.460i)6-s + (0.770 + 0.637i)7-s + (0.553 − 0.833i)8-s + (−0.844 − 0.536i)9-s + (−0.756 + 0.653i)10-s + (0.931 − 0.363i)11-s + (0.623 − 0.781i)12-s + (0.0515 − 0.998i)13-s + (−0.774 + 0.632i)14-s + (−0.810 + 0.585i)15-s + (0.709 + 0.704i)16-s + (−0.701 + 0.713i)17-s + ⋯
L(s)  = 1  + (−0.194 + 0.980i)2-s + (−0.279 + 0.960i)3-s + (−0.924 − 0.380i)4-s + (0.788 + 0.615i)5-s + (−0.887 − 0.460i)6-s + (0.770 + 0.637i)7-s + (0.553 − 0.833i)8-s + (−0.844 − 0.536i)9-s + (−0.756 + 0.653i)10-s + (0.931 − 0.363i)11-s + (0.623 − 0.781i)12-s + (0.0515 − 0.998i)13-s + (−0.774 + 0.632i)14-s + (−0.810 + 0.585i)15-s + (0.709 + 0.704i)16-s + (−0.701 + 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.783 + 0.621i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.783 + 0.621i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7980067648 + 2.288177692i\)
\(L(\frac12)\) \(\approx\) \(0.7980067648 + 2.288177692i\)
\(L(1)\) \(\approx\) \(0.7516551039 + 0.8413893047i\)
\(L(1)\) \(\approx\) \(0.7516551039 + 0.8413893047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.194 + 0.980i)T \)
3 \( 1 + (-0.279 + 0.960i)T \)
5 \( 1 + (0.788 + 0.615i)T \)
7 \( 1 + (0.770 + 0.637i)T \)
11 \( 1 + (0.931 - 0.363i)T \)
13 \( 1 + (0.0515 - 0.998i)T \)
17 \( 1 + (-0.701 + 0.713i)T \)
19 \( 1 + (0.754 + 0.656i)T \)
23 \( 1 + (0.986 - 0.161i)T \)
29 \( 1 + (-0.739 - 0.672i)T \)
31 \( 1 + (-0.317 - 0.948i)T \)
37 \( 1 + (0.837 + 0.546i)T \)
41 \( 1 + (-0.926 - 0.375i)T \)
43 \( 1 + (-0.992 + 0.121i)T \)
47 \( 1 + (0.830 - 0.557i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.952 - 0.304i)T \)
61 \( 1 + (0.959 + 0.280i)T \)
67 \( 1 + (0.416 - 0.909i)T \)
71 \( 1 + (0.320 - 0.947i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (-0.932 + 0.360i)T \)
83 \( 1 + (-0.988 + 0.152i)T \)
89 \( 1 + (0.436 + 0.899i)T \)
97 \( 1 + (-0.664 - 0.747i)T \)
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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

−19.67074206848149681671058586565, −18.621935112394355928725422281904, −18.08290174983621221438929448512, −17.39233909255625732039866592342, −17.01889823164713386865627785861, −16.275072627819612066955510457879, −14.48550238937861730296864925474, −14.163374098187882223237339166677, −13.31026586665949179620108420956, −12.99164545272335064842030057044, −11.83249709003988636642864266140, −11.5511815770274654248937815707, −10.79722150929439894286629696491, −9.7286037969105439029791164918, −8.96195670596412883614021073786, −8.56463437249320195221176940198, −7.19204868392371322489029756035, −6.935412543989827878247867791819, −5.49643762893751647043108436559, −4.8757431299654896951982188579, −4.11563364551570686470311698939, −2.81101509939438846569313470805, −1.8711674065312574613273107269, −1.3463843002696838734111294686, −0.69179665845607860557847036594, 0.73608575379942466339614105685, 1.93509764777937565467788262963, 3.23045704512382911416799999799, 4.02792294273528928223084723611, 5.06788908161030431971958919762, 5.683429824583405990483133649842, 6.14028905294576904718164568648, 7.0931278898065423965463098038, 8.228493066248169045982250770577, 8.783982967900243368002811308772, 9.593949330307658784648522189382, 10.163167893277139906924340138976, 11.033884304556805016262182074132, 11.66180643254717978900797439595, 12.9403880668489461954008391093, 13.738908198945819519095286975, 14.56338985100652130132998037295, 15.10209436725527621837470975581, 15.33830097594360295642143417080, 16.641480403556197571694620235399, 17.01564651791804598246698312308, 17.70062835644705950930448364291, 18.300403320238287047198123856292, 19.02053170788757150485179588347, 20.14808265683144428222834482103

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