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Properties

Label 1-1045-1045.37-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.837 - 0.545i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (−0.587 − 0.809i)18-s − 21-s − i·23-s + (−0.309 − 0.951i)24-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − i·12-s + (−0.951 + 0.309i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (−0.587 − 0.809i)18-s − 21-s − i·23-s + (−0.309 − 0.951i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.837 - 0.545i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.837 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8196550589 - 2.760065445i\)
\(L(\frac12)\) \(\approx\) \(0.8196550589 - 2.760065445i\)
\(L(1)\) \(\approx\) \(1.518969173 - 1.292283279i\)
\(L(1)\) \(\approx\) \(1.518969173 - 1.292283279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−21.8630780839690300220978644309, −21.26757363305766415988887231440, −20.56369671979833960507455358476, −19.60191900604120805365703582331, −19.17008884129210309395584446535, −17.77316919560566501127919232141, −16.804919728267749631779847192380, −16.083252077372662386163913162284, −15.53422943314453140327582945006, −14.71701975042995263498518921271, −14.26241459961755384052988496469, −13.22075172847370134522464849300, −12.5063919033838781116908575147, −11.696277319626180513026127286, −10.74837195287528392923171595049, −9.7191220248689480539769735871, −9.111332554818979054384257913150, −7.893112862048283464579000757242, −7.3604916843661417691035563923, −5.96797196557910901119681435005, −5.39526730085591938764059342279, −4.5421675530494491436323412797, −3.4652612627560238439565794016, −2.90242799658095041203505123216, −1.998417972805517532461621312818, 0.75677752459343166378909366639, 1.88491358234420071253531288167, 2.80148994348870401926312997289, 3.61009542318800367159670480611, 4.45891268181837633276106106837, 5.683739782615391372425872481791, 6.54918877264244375831244439354, 7.24972551857564788377367541160, 7.93382436031955762715622726740, 9.35151396926867081496144280853, 10.02937414664271471896716417731, 11.00738797761297677022232414805, 12.03338418523844062189648200192, 12.65763009850331153974167238718, 13.22837339150553860566652085628, 14.15157553498441690911696683714, 14.5578261615403308916832418424, 15.41726465225467089481723569694, 16.638080231705663280874383031964, 17.05875376999360162681667040004, 18.62177645176403811962580749910, 18.93397983256259610647448090474, 20.0480152774453119416732401123, 20.164370189628898805171126784329, 21.15581861735395640753869435083

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