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Properties

Label 1-3971-3971.987-r0-0-0
Degree $1$
Conductor $3971$
Sign $-0.920 + 0.391i$
Analytic cond. $18.4412$
Root an. cond. $18.4412$
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  + (0.180 + 0.983i)2-s + (−0.894 + 0.446i)3-s + (−0.934 + 0.355i)4-s + (0.746 − 0.665i)5-s + (−0.601 − 0.799i)6-s + (−0.0495 + 0.998i)7-s + (−0.518 − 0.854i)8-s + (0.601 − 0.799i)9-s + (0.789 + 0.614i)10-s + (0.677 − 0.735i)12-s + (−0.461 + 0.887i)13-s + (−0.991 + 0.131i)14-s + (−0.371 + 0.928i)15-s + (0.746 − 0.665i)16-s + (0.627 − 0.778i)17-s + (0.894 + 0.446i)18-s + ⋯
L(s)  = 1  + (0.180 + 0.983i)2-s + (−0.894 + 0.446i)3-s + (−0.934 + 0.355i)4-s + (0.746 − 0.665i)5-s + (−0.601 − 0.799i)6-s + (−0.0495 + 0.998i)7-s + (−0.518 − 0.854i)8-s + (0.601 − 0.799i)9-s + (0.789 + 0.614i)10-s + (0.677 − 0.735i)12-s + (−0.461 + 0.887i)13-s + (−0.991 + 0.131i)14-s + (−0.371 + 0.928i)15-s + (0.746 − 0.665i)16-s + (0.627 − 0.778i)17-s + (0.894 + 0.446i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3971\)    =    \(11 \cdot 19^{2}\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(18.4412\)
Root analytic conductor: \(18.4412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3971} (987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3971,\ (0:\ ),\ -0.920 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2115718458 + 1.038009911i\)
\(L(\frac12)\) \(\approx\) \(0.2115718458 + 1.038009911i\)
\(L(1)\) \(\approx\) \(0.6616849349 + 0.5312527673i\)
\(L(1)\) \(\approx\) \(0.6616849349 + 0.5312527673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.180 + 0.983i)T \)
3 \( 1 + (-0.894 + 0.446i)T \)
5 \( 1 + (0.746 - 0.665i)T \)
7 \( 1 + (-0.0495 + 0.998i)T \)
13 \( 1 + (-0.461 + 0.887i)T \)
17 \( 1 + (0.627 - 0.778i)T \)
23 \( 1 + (-0.986 + 0.164i)T \)
29 \( 1 + (0.0495 - 0.998i)T \)
31 \( 1 + (0.724 - 0.689i)T \)
37 \( 1 + (0.956 + 0.293i)T \)
41 \( 1 + (-0.768 - 0.639i)T \)
43 \( 1 + (-0.789 + 0.614i)T \)
47 \( 1 + (-0.574 + 0.818i)T \)
53 \( 1 + (0.0165 + 0.999i)T \)
59 \( 1 + (-0.371 + 0.928i)T \)
61 \( 1 + (-0.922 + 0.386i)T \)
67 \( 1 + (0.0825 - 0.996i)T \)
71 \( 1 + (0.973 - 0.229i)T \)
73 \( 1 + (-0.0495 + 0.998i)T \)
79 \( 1 + (-0.340 - 0.940i)T \)
83 \( 1 + (0.995 + 0.0990i)T \)
89 \( 1 + (-0.546 + 0.837i)T \)
97 \( 1 + (0.973 + 0.229i)T \)
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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

−18.183345180957895622168109330046, −17.704185055015060383674802921154, −17.077348071663082916313216664219, −16.53778958546270155619101801020, −15.32839620065159255769772365132, −14.44454568584971571362954620537, −13.98815080716567123415700839950, −13.149230276614058550583315651667, −12.803406061264191726918054360731, −11.99112922848820044624633490713, −11.24825670729177462703805185464, −10.51035244242002380305303602425, −10.19435276748325982504468673745, −9.74341402693914788094191917932, −8.36202679068744539859533149195, −7.70322229957762979027215339878, −6.74883119700638194305104335607, −6.14851529181198081337782334612, −5.32330675038331236217138862502, −4.76524247168132858655777935547, −3.692216168348063845896810874495, −3.05534707142583035807548829374, −1.973827284507643756681269963092, −1.40950706631862442340266637822, −0.413592072002425254829454279530, 0.83963288540946481278885711725, 2.002522191025022415711780770522, 3.092065975400688593531559996802, 4.32708356761858361037879590435, 4.71190073842883911943242707874, 5.45997146956857478633025166810, 6.0838811381361586470075212296, 6.43719714604258208204879575084, 7.54250053088272176218280646623, 8.32705839808657288463448091574, 9.2942605374162775134445665384, 9.55483521565481980722600104168, 10.14663335812208726567617102717, 11.55436539810054040677203206787, 12.02790563467851517373450085208, 12.54363719321611046989399883020, 13.476598173265086090690403982033, 14.029166259998971449428958137791, 14.92708331662285710121966683160, 15.50552093004975188257389800041, 16.22113975467568866576854286179, 16.68903548990954822972319459498, 17.198796120429886726101856538085, 17.99184192717270656205979983159, 18.41632684309746718919107351394

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