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Properties

Label 1-1037-1037.26-r1-0-0
Degree $1$
Conductor $1037$
Sign $-0.971 + 0.238i$
Analytic cond. $111.441$
Root an. cond. $111.441$
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  + (−0.913 + 0.406i)2-s + (0.156 + 0.987i)3-s + (0.669 − 0.743i)4-s + (−0.544 − 0.838i)5-s + (−0.544 − 0.838i)6-s + (0.629 − 0.777i)7-s + (−0.309 + 0.951i)8-s + (−0.951 + 0.309i)9-s + (0.838 + 0.544i)10-s + (0.707 + 0.707i)11-s + (0.838 + 0.544i)12-s + (0.5 − 0.866i)13-s + (−0.258 + 0.965i)14-s + (0.743 − 0.669i)15-s + (−0.104 − 0.994i)16-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.156 + 0.987i)3-s + (0.669 − 0.743i)4-s + (−0.544 − 0.838i)5-s + (−0.544 − 0.838i)6-s + (0.629 − 0.777i)7-s + (−0.309 + 0.951i)8-s + (−0.951 + 0.309i)9-s + (0.838 + 0.544i)10-s + (0.707 + 0.707i)11-s + (0.838 + 0.544i)12-s + (0.5 − 0.866i)13-s + (−0.258 + 0.965i)14-s + (0.743 − 0.669i)15-s + (−0.104 − 0.994i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(111.441\)
Root analytic conductor: \(111.441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (1:\ ),\ -0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03519451934 + 0.2904390497i\)
\(L(\frac12)\) \(\approx\) \(0.03519451934 + 0.2904390497i\)
\(L(1)\) \(\approx\) \(0.6228892298 + 0.1418869194i\)
\(L(1)\) \(\approx\) \(0.6228892298 + 0.1418869194i\)

Euler product

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

−20.89519771892260783719071949459, −19.86155699768239684379254280733, −19.22020274730356535962960516501, −18.61753993558652147102382461861, −18.33987538786352130463495302352, −17.26193204140201853606174669712, −16.65295024758928200592615792131, −15.44679513806172132929893730096, −14.76445125699994442418010890533, −13.90675110827290739811320130340, −12.85671176378373300510058873447, −11.95305077369988361580590797601, −11.25275348196366365863188303012, −11.10336461846690729947877982273, −9.53995833345263344563048611710, −8.66475563282043307931669627368, −8.22979310769990659104578422218, −7.21373383374459533688194257581, −6.60849270568239490581622883418, −5.75152925671016473227977183042, −3.92262270961209039476665349442, −3.154090086930893673742330861400, −2.08790287804362530338097103618, −1.47677751304917029753902610011, −0.09384841211449350403866242315, 0.928264041473529129240785033335, 2.036694929692077285171820146754, 3.56028030942594602386790924296, 4.4469455216840460372302101210, 5.136201026996532449428135105724, 6.217913281848219123827820866758, 7.43843020282815614063052262637, 8.08825688811895645960563578030, 8.8936296187600539049933885232, 9.44357229470008422229809558494, 10.64747670939194422357785475974, 10.879426833337706801358472053153, 11.93008844391684431459342333584, 13.00877389601707635172986499668, 14.26877455309860126227163664007, 14.95136408230663455066077501408, 15.479029541166531927659217079971, 16.46070502251236706928126184673, 16.95868244847609575186977405058, 17.468422580533688857932726440316, 18.5465118240730663555998185170, 19.74294382445674251335975020884, 20.112252661076846283244991494564, 20.625599213479525748466788288587, 21.383814669415573721730180485245

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