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Properties

Label 1-1037-1037.718-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.987 + 0.159i$
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  + (0.5 − 0.866i)2-s − i·3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s − 8-s − 9-s + (0.866 − 0.5i)10-s + i·11-s + (−0.866 + 0.5i)12-s + (−0.5 + 0.866i)13-s + (−0.866 + 0.5i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − i·3-s + (−0.5 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 − 0.5i)6-s + (−0.866 − 0.5i)7-s − 8-s − 9-s + (0.866 − 0.5i)10-s + i·11-s + (−0.866 + 0.5i)12-s + (−0.5 + 0.866i)13-s + (−0.866 + 0.5i)14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9601149200 + 0.07729435659i\)
\(L(\frac12)\) \(\approx\) \(0.9601149200 + 0.07729435659i\)
\(L(1)\) \(\approx\) \(0.9166093016 - 0.5277154636i\)
\(L(1)\) \(\approx\) \(0.9166093016 - 0.5277154636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + iT \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−21.91322596879151597763668641182, −20.93385629207089286901143403636, −20.35986848999056602316915267145, −19.20676955007643948311769130891, −18.11512985636869589433827192189, −17.324370863988274921968606908679, −16.56852579214757747042098557580, −16.14904247721988993826942494988, −15.31372369782101871273444340736, −14.61542047209962465613012773986, −13.68739736175108039398468201879, −13.069577009039863190741451132319, −12.265686787920620539747720336413, −11.158711122320201539684847857259, −10.05516885751021378842784723256, −9.27502891767227762014765633480, −8.81376099672682661787340791907, −7.828441026956884567570843821527, −6.452232008696774289856974674614, −5.788942691595932289037020905926, −5.25651191224081208107561913403, −4.336909313168575401078260507993, −3.16453507257319188830334539622, −2.65077469115826478137493468039, −0.31964232284963858489853270527, 1.48064291056942079233980296248, 1.98248308463314676202544529852, 3.009452831261223495702082481561, 3.848544406415901274557267386, 5.22029939664889423048866526087, 5.953593978165690051165435658245, 6.88431008716107791825873733854, 7.40641473440555904611788712535, 9.096564008381326940866869644227, 9.63362136797733967304867095157, 10.404905955615191131662552465551, 11.37400359280442651467354423474, 12.20894803327433868991695989608, 12.944008816802967520499603914822, 13.441541699188860372959171878415, 14.38787635807592593899411314263, 14.6575581342833055574086363317, 16.17133105419045859212969119603, 17.24851118763597397344747532595, 17.85904499152091096232301607468, 18.68446413659118272549490812414, 19.21723542896519379207895128952, 20.068273701829686336779840534023, 20.6160229055877021887490244952, 21.77757391655700650999609145810

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