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Properties

Label 1-1037-1037.150-r0-0-0
Degree $1$
Conductor $1037$
Sign $-0.932 - 0.360i$
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  + (−0.156 − 0.987i)2-s + (−0.522 − 0.852i)3-s + (−0.951 + 0.309i)4-s + (−0.996 − 0.0784i)5-s + (−0.760 + 0.649i)6-s + (0.522 − 0.852i)7-s + (0.453 + 0.891i)8-s + (−0.453 + 0.891i)9-s + (0.0784 + 0.996i)10-s + (−0.382 − 0.923i)11-s + (0.760 + 0.649i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.522 − 0.852i)3-s + (−0.951 + 0.309i)4-s + (−0.996 − 0.0784i)5-s + (−0.760 + 0.649i)6-s + (0.522 − 0.852i)7-s + (0.453 + 0.891i)8-s + (−0.453 + 0.891i)9-s + (0.0784 + 0.996i)10-s + (−0.382 − 0.923i)11-s + (0.760 + 0.649i)12-s + i·13-s + (−0.923 − 0.382i)14-s + (0.453 + 0.891i)15-s + (0.809 − 0.587i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ -0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1530677705 - 0.8214973698i\)
\(L(\frac12)\) \(\approx\) \(0.1530677705 - 0.8214973698i\)
\(L(1)\) \(\approx\) \(0.4719931686 - 0.5284170073i\)
\(L(1)\) \(\approx\) \(0.4719931686 - 0.5284170073i\)

Euler product

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

−22.26862764929330378522994801358, −21.23196462664769700471638639657, −20.4148516355519187697774503219, −19.50453357780224514213607988826, −18.42098754919725348508838563474, −17.89849182653043962386773257384, −17.18291635260628515547974463958, −16.20372984641137239892220889746, −15.467482124736460751598172727564, −15.22635419015560316681329791579, −14.58357866643902886337039618045, −13.2632654970011922881217668178, −12.226274788908676939656812183194, −11.63154903273800464022519764819, −10.55514921023072439313532432501, −9.800579185236006783974289775748, −8.92366563503038550577394829920, −8.05136930721501348514628871477, −7.42296195906537877197795213573, −6.292757307728278302584333640094, −5.261366454407269770916939435048, −4.91059390600793415014070354069, −3.912389189844740468765161962947, −2.87456883693361406846436245251, −0.89272911749898654170547959496, 0.63769684759425336005578469391, 1.28671823986827524200920432831, 2.61203714421052105948077633826, 3.5762253775004045069791996864, 4.58931868001169445532923954282, 5.26055981090609087387706169013, 6.750403935420778139571691312075, 7.50365507097528476887159832147, 8.278319274673209124913015423459, 8.97365114263555143984159962612, 10.47559621693869222793042614243, 10.98228006335082357560835144888, 11.636650583664812562160655949073, 12.23480996880150815400170356324, 13.16323390647354483220108650402, 13.905998197881607405291724609462, 14.48396789889850662161239127138, 16.21392694929473899660294740840, 16.52371641216729207777104155681, 17.622997028210593885844330426887, 18.19477599467491574310155991553, 19.176372412619710873652283721064, 19.38710416883169164123468397795, 20.321642817802683666820980307197, 21.09963635014182478176790177171

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