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Properties

Label 1-1037-1037.117-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.945 + 0.324i$
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  + (0.743 + 0.669i)2-s + (−0.891 − 0.453i)3-s + (0.104 + 0.994i)4-s + (0.933 − 0.358i)5-s + (−0.358 − 0.933i)6-s + (−0.838 − 0.544i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.933 + 0.358i)10-s + (−0.707 − 0.707i)11-s + (0.358 − 0.933i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.891 − 0.453i)3-s + (0.104 + 0.994i)4-s + (0.933 − 0.358i)5-s + (−0.358 − 0.933i)6-s + (−0.838 − 0.544i)7-s + (−0.587 + 0.809i)8-s + (0.587 + 0.809i)9-s + (0.933 + 0.358i)10-s + (−0.707 − 0.707i)11-s + (0.358 − 0.933i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.994 − 0.104i)15-s + (−0.978 + 0.207i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.718182927 + 0.2865632419i\)
\(L(\frac12)\) \(\approx\) \(1.718182927 + 0.2865632419i\)
\(L(1)\) \(\approx\) \(1.260351256 + 0.2395672126i\)
\(L(1)\) \(\approx\) \(1.260351256 + 0.2395672126i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (-0.891 - 0.453i)T \)
5 \( 1 + (0.933 - 0.358i)T \)
7 \( 1 + (-0.838 - 0.544i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.207 - 0.978i)T \)
23 \( 1 + (0.987 + 0.156i)T \)
29 \( 1 + (0.258 + 0.965i)T \)
31 \( 1 + (-0.998 - 0.0523i)T \)
37 \( 1 + (0.453 + 0.891i)T \)
41 \( 1 + (0.891 - 0.453i)T \)
43 \( 1 + (0.994 + 0.104i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (-0.358 + 0.933i)T \)
73 \( 1 + (0.358 - 0.933i)T \)
79 \( 1 + (0.777 - 0.629i)T \)
83 \( 1 + (0.743 + 0.669i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.998 + 0.0523i)T \)
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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

−21.52412979393543239520814358216, −20.993832632095977639444003081545, −20.33726454939727769501929286285, −19.11854780960888576028281888633, −18.31387234084077748912369087268, −17.90584667447570600013295228795, −16.77188494597814886548047597382, −15.81836234436150195129236992156, −15.25372762879404413803545240503, −14.45526979566606207002243529043, −13.290680044155036553438899883125, −12.73016834711607319130445951598, −12.202986346289819968552336892014, −10.89471684917488756174457869173, −10.58058160044005120954324770637, −9.68557291587705951501213655275, −9.23245364050246765256731226558, −7.39044104873449525148877084303, −6.2743250937998479787743017739, −5.791065357775137546239764382464, −5.2047694647450947065891954108, −4.06634827342548997418721014450, −3.065802947203134225544997903783, −2.26456829379188077881693401646, −0.9514632330455501551793431331, 0.831302108761595192286138145923, 2.24407171337232487728693853482, 3.31413228619406601034136371660, 4.53378391548606608179945313930, 5.31392214083222480099786321873, 6.0123621776402342082939973375, 6.756649474415896293432258157274, 7.32430107233192711651663309124, 8.62112032454476742322155776327, 9.42893409963430890321601964167, 10.692150787768223154537270735931, 11.26428789235968479578463612283, 12.447398920409357642691494991784, 13.0997349307573459082121258786, 13.50172910967886982649316576101, 14.20082164683570372733397552865, 15.58398947672334735599263394673, 16.36362650482564827518680777511, 16.66048301256343390301491583666, 17.501103636642092002036214129023, 18.23437324376620757454616944331, 19.079934354563441647101556974704, 20.25836818344980490245133800016, 21.209525950784442147596727531202, 21.78759686161278996135986830027

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