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Properties

Label 1-1037-1037.114-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.998 - 0.0457i$
Analytic cond. $4.81580$
Root an. cond. $4.81580$
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  + (0.891 − 0.453i)2-s + (0.760 + 0.649i)3-s + (0.587 − 0.809i)4-s + (0.522 + 0.852i)5-s + (0.972 + 0.233i)6-s + (0.760 − 0.649i)7-s + (0.156 − 0.987i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)10-s + (0.923 − 0.382i)11-s + (0.972 − 0.233i)12-s + i·13-s + (0.382 − 0.923i)14-s + (−0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (0.760 + 0.649i)3-s + (0.587 − 0.809i)4-s + (0.522 + 0.852i)5-s + (0.972 + 0.233i)6-s + (0.760 − 0.649i)7-s + (0.156 − 0.987i)8-s + (0.156 + 0.987i)9-s + (0.852 + 0.522i)10-s + (0.923 − 0.382i)11-s + (0.972 − 0.233i)12-s + i·13-s + (0.382 − 0.923i)14-s + (−0.156 + 0.987i)15-s + (−0.309 − 0.951i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.998 - 0.0457i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.998 - 0.0457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.196284694 - 0.09610282262i\)
\(L(\frac12)\) \(\approx\) \(4.196284694 - 0.09610282262i\)
\(L(1)\) \(\approx\) \(2.614444788 - 0.09811349123i\)
\(L(1)\) \(\approx\) \(2.614444788 - 0.09811349123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 + (0.760 + 0.649i)T \)
5 \( 1 + (0.522 + 0.852i)T \)
7 \( 1 + (0.760 - 0.649i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.453 - 0.891i)T \)
23 \( 1 + (-0.972 - 0.233i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.0784 - 0.996i)T \)
37 \( 1 + (-0.0784 + 0.996i)T \)
41 \( 1 + (-0.649 - 0.760i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.972 + 0.233i)T \)
73 \( 1 + (-0.972 + 0.233i)T \)
79 \( 1 + (0.522 - 0.852i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.0784 + 0.996i)T \)
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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

−21.55812193894572547381323509325, −20.77870081990001507544156640514, −20.216779253019375328466842970604, −19.57008500676707413839470608964, −18.13437738454971991448367024388, −17.71529417522629769194877207234, −16.87988073921766266844449208875, −15.865839720132350349670498230043, −15.042554056255056836842810989273, −14.34695815699386896186245953979, −13.85403943899005727706241675877, −12.786517114147177065889416139869, −12.35540298222122919172144239765, −11.773267895449093130619369199105, −10.334588457646945240759911613149, −9.11874198198022257929458047525, −8.41382199582539139204312545154, −7.889219352050032491673665186262, −6.7808089649589516459498619652, −5.91578209424606433387656632031, −5.18035963032012680053098121486, −4.172151529217818735659148515010, −3.22460303060777134394086028604, −2.022589675295158801536942862329, −1.51449622586999298850002790659, 1.5198561605003809535524286133, 2.27432081456357607961182814753, 3.200970640685717127346799856623, 4.20935736463417224030721755430, 4.55021400884888362147547490777, 5.93717103918316213985817073372, 6.70465249692645724730736339246, 7.63226759960035726243829681830, 8.85263625329188659970479870217, 9.77817154233911197339846725904, 10.38284304187205817126533862524, 11.300113628932974893480512547057, 11.69106108442101229767681803884, 13.35273599010761756010825096582, 13.72828273657754478808592801842, 14.47508070347107440731273467306, 14.80034397662733469979833285532, 15.79451069125959112798815721407, 16.74391741306173407258401106303, 17.61072625373415480522156361932, 18.94062327418915203419289477004, 19.27238164111409979527049339739, 20.26884419297289115930008826067, 20.86607910644883460638926142672, 21.66269227700596850314267887382

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