Properties

Label 1-2011-2011.40-r1-0-0
Degree $1$
Conductor $2011$
Sign $-0.193 + 0.981i$
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  + (−0.828 + 0.559i)2-s + (0.707 − 0.706i)3-s + (0.373 − 0.927i)4-s + (0.696 + 0.717i)5-s + (−0.191 + 0.981i)6-s + (−0.586 + 0.809i)7-s + (0.209 + 0.977i)8-s + (0.00156 − 0.999i)9-s + (−0.978 − 0.204i)10-s + (−0.999 − 0.0250i)11-s + (−0.390 − 0.920i)12-s + (0.998 − 0.0562i)13-s + (0.0328 − 0.999i)14-s + (0.999 + 0.0156i)15-s + (−0.720 − 0.693i)16-s + (−0.491 − 0.870i)17-s + ⋯
L(s)  = 1  + (−0.828 + 0.559i)2-s + (0.707 − 0.706i)3-s + (0.373 − 0.927i)4-s + (0.696 + 0.717i)5-s + (−0.191 + 0.981i)6-s + (−0.586 + 0.809i)7-s + (0.209 + 0.977i)8-s + (0.00156 − 0.999i)9-s + (−0.978 − 0.204i)10-s + (−0.999 − 0.0250i)11-s + (−0.390 − 0.920i)12-s + (0.998 − 0.0562i)13-s + (0.0328 − 0.999i)14-s + (0.999 + 0.0156i)15-s + (−0.720 − 0.693i)16-s + (−0.491 − 0.870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $-0.193 + 0.981i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ -0.193 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8156892698 + 0.9919475917i\)
\(L(\frac12)\) \(\approx\) \(0.8156892698 + 0.9919475917i\)
\(L(1)\) \(\approx\) \(0.8796147707 + 0.1545291024i\)
\(L(1)\) \(\approx\) \(0.8796147707 + 0.1545291024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (-0.828 + 0.559i)T \)
3 \( 1 + (0.707 - 0.706i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
7 \( 1 + (-0.586 + 0.809i)T \)
11 \( 1 + (-0.999 - 0.0250i)T \)
13 \( 1 + (0.998 - 0.0562i)T \)
17 \( 1 + (-0.491 - 0.870i)T \)
19 \( 1 + (0.123 - 0.992i)T \)
23 \( 1 + (-0.129 + 0.991i)T \)
29 \( 1 + (0.758 - 0.651i)T \)
31 \( 1 + (-0.353 - 0.935i)T \)
37 \( 1 + (-0.968 + 0.247i)T \)
41 \( 1 + (0.273 + 0.961i)T \)
43 \( 1 + (0.291 + 0.956i)T \)
47 \( 1 + (-0.480 + 0.876i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.943 - 0.331i)T \)
61 \( 1 + (0.986 + 0.164i)T \)
67 \( 1 + (0.837 + 0.546i)T \)
71 \( 1 + (-0.770 - 0.637i)T \)
73 \( 1 + (0.197 + 0.980i)T \)
79 \( 1 + (0.347 - 0.937i)T \)
83 \( 1 + (0.436 + 0.899i)T \)
89 \( 1 + (-0.252 + 0.967i)T \)
97 \( 1 + (0.472 + 0.881i)T \)
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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

−19.75617153405638376322148096549, −18.90565209647582628844188889585, −18.19786339516383771520160613662, −17.31278801774728469672224014386, −16.62799267891815019202728303420, −16.04448828959566550973060895200, −15.60411175950984654492922485247, −14.18100384068613982725824640022, −13.64142220755222568667269913644, −12.84635698311574129054147008184, −12.39241663516162641421769115437, −10.85227988332102471952634729919, −10.391945746306223372304587201369, −10.107020045346124720489008212961, −8.93811695031319704134472296145, −8.64656991268020708135070526589, −7.88347189497387093213895862573, −6.87293499171238304584738396128, −5.870550802092270250474721164589, −4.74493303382071957040019455595, −3.87173067515527766698914644948, −3.25764972616840747475405502655, −2.19203420966586769140549703106, −1.47160244444318165884424355823, −0.2992206493661552941702239417, 0.84825030367611843127919484976, 1.99330800266442793061735957951, 2.60672293548767130284007326179, 3.25741208603707303882156606623, 5.00280649640894489456993169072, 5.9544822627070812657766462375, 6.41202775395357224283129881155, 7.178547945926611057341635356944, 7.94710529606788587414994579502, 8.71679730221300430409355960452, 9.50057785233477026264801230146, 9.85230070580152297802984326661, 11.07587270747159935745982785465, 11.571772525018455165776584537814, 12.976293174869957316608932481155, 13.450905549365293718469749381, 14.08737547102598644468618007083, 15.0531563147193586169887845760, 15.606399147588644043417500700723, 16.07826591104385869377883564799, 17.47462425532068962620044555930, 17.99831041498334759351779527458, 18.34698847590637929733124617637, 19.09971824870924000447255699353, 19.56293434756709461813236743862

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