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Properties

Label 1-1081-1081.850-r1-0-0
Degree $1$
Conductor $1081$
Sign $0.859 - 0.510i$
Analytic cond. $116.169$
Root an. cond. $116.169$
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 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.203 + 0.979i)5-s + (−0.0682 − 0.997i)6-s + (−0.962 − 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.682 + 0.730i)10-s + (0.990 − 0.136i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (−0.682 − 0.730i)14-s + (0.917 − 0.398i)15-s + (−0.576 + 0.816i)16-s + (0.990 + 0.136i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (−0.576 − 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.203 + 0.979i)5-s + (−0.0682 − 0.997i)6-s + (−0.962 − 0.269i)7-s + (−0.0682 + 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.682 + 0.730i)10-s + (0.990 − 0.136i)11-s + (0.460 − 0.887i)12-s + (−0.775 − 0.631i)13-s + (−0.682 − 0.730i)14-s + (0.917 − 0.398i)15-s + (−0.576 + 0.816i)16-s + (0.990 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1081\)    =    \(23 \cdot 47\)
Sign: $0.859 - 0.510i$
Analytic conductor: \(116.169\)
Root analytic conductor: \(116.169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1081} (850, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1081,\ (1:\ ),\ 0.859 - 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.492478508 - 0.4095999922i\)
\(L(\frac12)\) \(\approx\) \(1.492478508 - 0.4095999922i\)
\(L(1)\) \(\approx\) \(1.124402117 + 0.2337209619i\)
\(L(1)\) \(\approx\) \(1.124402117 + 0.2337209619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.854 + 0.519i)T \)
3 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (-0.203 + 0.979i)T \)
7 \( 1 + (-0.962 - 0.269i)T \)
11 \( 1 + (0.990 - 0.136i)T \)
13 \( 1 + (-0.775 - 0.631i)T \)
17 \( 1 + (0.990 + 0.136i)T \)
19 \( 1 + (-0.203 - 0.979i)T \)
29 \( 1 + (-0.775 + 0.631i)T \)
31 \( 1 + (-0.576 + 0.816i)T \)
37 \( 1 + (-0.682 + 0.730i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.460 - 0.887i)T \)
53 \( 1 + (0.0682 + 0.997i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (-0.682 - 0.730i)T \)
67 \( 1 + (-0.962 + 0.269i)T \)
71 \( 1 + (0.854 - 0.519i)T \)
73 \( 1 + (-0.334 - 0.942i)T \)
79 \( 1 + (0.917 - 0.398i)T \)
83 \( 1 + (0.990 - 0.136i)T \)
89 \( 1 + (-0.203 + 0.979i)T \)
97 \( 1 + (0.576 + 0.816i)T \)
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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

−21.251531622744456288392197294179, −20.86770004198678429494158937262, −19.83198310532579092691837197644, −19.41429041793526200743125997648, −18.41291146514856487832124914507, −16.86644489753386010350432090104, −16.66826059331893737817678679123, −15.92567286315199234534800536060, −14.936652869300102748340283041875, −14.43615385036898704871441341616, −13.233193933046558908124371202667, −12.38970387805972594223501818111, −11.9782819856626786431752875135, −11.317101573179761605233313890696, −9.914055988193488066340077468347, −9.72549351451223831468777937820, −8.890234746044867467236359042382, −7.33179929932488784842029930991, −6.196581316788519312957678346842, −5.67976922615955073794157343977, −4.74156588557847102499312282939, −3.96109646272027705590673943981, −3.39607362397210868249887375741, −1.95327875746668755827336351690, −0.78995344964126461624528925626, 0.321424718858172194525088674963, 1.9346834612868149551984088630, 3.07030120990069301117989644811, 3.59261686217746023135561095490, 4.9598302892755300381589498730, 5.84079877796692826619557449374, 6.63650546096479019576235944554, 7.10404423443286383244471702654, 7.74388390191982502745692819686, 9.00942377870438084566243324943, 10.34661485561779733933107833603, 11.02379072343991129966361806231, 12.089105905627935336319397711691, 12.39064909942069411924462122623, 13.41780865536713126878926904737, 14.0472102733095367942743639185, 14.82871460607387185463224363259, 15.63282583882239927386153037913, 16.64039030575116611385050096765, 17.124730804429803879695203518113, 17.92271774899850837002453578453, 18.96629371054579736315116259370, 19.54501196686411909286491544674, 20.29245448751853063809403018491, 21.84604953959859895916906886312

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