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Properties

Label 1-1037-1037.134-r0-0-0
Degree $1$
Conductor $1037$
Sign $-0.463 - 0.885i$
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 + (0.453 + 0.891i)3-s + (0.104 − 0.994i)4-s + (0.358 − 0.933i)5-s + (−0.933 − 0.358i)6-s + (−0.544 − 0.838i)7-s + (0.587 + 0.809i)8-s + (−0.587 + 0.809i)9-s + (0.358 + 0.933i)10-s + (−0.707 − 0.707i)11-s + (0.933 − 0.358i)12-s + (0.5 − 0.866i)13-s + (0.965 + 0.258i)14-s + (0.994 − 0.104i)15-s + (−0.978 − 0.207i)16-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.453 + 0.891i)3-s + (0.104 − 0.994i)4-s + (0.358 − 0.933i)5-s + (−0.933 − 0.358i)6-s + (−0.544 − 0.838i)7-s + (0.587 + 0.809i)8-s + (−0.587 + 0.809i)9-s + (0.358 + 0.933i)10-s + (−0.707 − 0.707i)11-s + (0.933 − 0.358i)12-s + (0.5 − 0.866i)13-s + (0.965 + 0.258i)14-s + (0.994 − 0.104i)15-s + (−0.978 − 0.207i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.463 - 0.885i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ -0.463 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1929498693 - 0.3188608526i\)
\(L(\frac12)\) \(\approx\) \(0.1929498693 - 0.3188608526i\)
\(L(1)\) \(\approx\) \(0.6617340387 + 0.08138380446i\)
\(L(1)\) \(\approx\) \(0.6617340387 + 0.08138380446i\)

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 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (0.358 - 0.933i)T \)
7 \( 1 + (-0.544 - 0.838i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.207 - 0.978i)T \)
23 \( 1 + (0.156 + 0.987i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (0.0523 + 0.998i)T \)
37 \( 1 + (-0.891 - 0.453i)T \)
41 \( 1 + (-0.453 + 0.891i)T \)
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 \( 1 + (-0.933 + 0.358i)T \)
73 \( 1 + (0.933 - 0.358i)T \)
79 \( 1 + (-0.629 + 0.777i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.0523 - 0.998i)T \)
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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

−21.669444415408370627471851453287, −20.79616159143651209311950678236, −20.26924605731727046463511982580, −19.005639715788747937142929969920, −18.71138192074561436576585711406, −18.42413274673918891705630926490, −17.45534500414973373077577340371, −16.5957306402800566490192400173, −15.46633083459516160914949607810, −14.72182039094880487568093252143, −13.682765023640247853012004909605, −12.99479645848003586416293694630, −12.21953611938080769322251594340, −11.5653263162805185113163878544, −10.51042117311408633925670736246, −9.77126205021960842394836617117, −8.940219200356157507973502428345, −8.203182776476614535245003441288, −7.19348108364388346555261343438, −6.6461105597868754551862773298, −5.685599450574850681873090650791, −3.919491307460106990837800770584, −3.042773067423498187414047656733, −2.20803407027907318090944593475, −1.74616363444388752109212541892, 0.1797485169368670547954493957, 1.42326680738608257630630353185, 2.84946779478696328330441241493, 3.88501745077854441583448636122, 5.10078275123544718901768892242, 5.48476443775406430354573459656, 6.65411457258388133814072356418, 7.84165595061321264635075951264, 8.38425677775368196609184422034, 9.236839253955477569899290113, 9.84911193986797816318301704251, 10.638428081391335997118498403143, 11.25969690814159202425102756557, 13.05146817445301768169676343906, 13.475888728386585252649892054947, 14.26973253583922763820421413975, 15.51880624386779391545246705329, 15.79266759700112799582989389065, 16.569735292928268167077270413805, 17.20094214146581651764473087692, 17.92015998880396502059350778904, 19.16237033076929865770164976509, 19.77497662306802412458530519178, 20.41512378009888699871047774521, 21.05888749407683628411750742221

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