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Properties

Label 1-1045-1045.1032-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.366 + 0.930i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (0.743 + 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (−0.0348 − 0.999i)14-s + (−0.997 − 0.0697i)16-s + (−0.898 + 0.438i)17-s + (0.951 − 0.309i)18-s + (0.173 − 0.984i)21-s + (−0.342 + 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯
L(s)  = 1  + (−0.694 − 0.719i)2-s + (−0.529 − 0.848i)3-s + (−0.0348 + 0.999i)4-s + (−0.241 + 0.970i)6-s + (0.743 + 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.438 + 0.898i)9-s + (0.866 − 0.5i)12-s + (0.0697 + 0.997i)13-s + (−0.0348 − 0.999i)14-s + (−0.997 − 0.0697i)16-s + (−0.898 + 0.438i)17-s + (0.951 − 0.309i)18-s + (0.173 − 0.984i)21-s + (−0.342 + 0.939i)23-s + (−0.961 − 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1032, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.366 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2799409215 + 0.4109255728i\)
\(L(\frac12)\) \(\approx\) \(0.2799409215 + 0.4109255728i\)
\(L(1)\) \(\approx\) \(0.6023605141 - 0.1279969491i\)
\(L(1)\) \(\approx\) \(0.6023605141 - 0.1279969491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.694 - 0.719i)T \)
3 \( 1 + (-0.529 - 0.848i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (0.0697 + 0.997i)T \)
17 \( 1 + (-0.898 + 0.438i)T \)
23 \( 1 + (-0.342 + 0.939i)T \)
29 \( 1 + (0.882 - 0.469i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.848 - 0.529i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.139 + 0.990i)T \)
53 \( 1 + (0.829 - 0.559i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.559 - 0.829i)T \)
73 \( 1 + (0.927 + 0.374i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.694 - 0.719i)T \)
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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

−20.89547176713075632002073424749, −20.31243459798332463405035976813, −19.69375251123057084823437473282, −18.30992546955080316716850623270, −17.8582446976360516665466479412, −17.17236205441352952727322280076, −16.49259547529098102874828204193, −15.69641399478041867163605400432, −15.07363373617651587818943163572, −14.31035989683406282606430767364, −13.4819067828823281281315579900, −12.150572631795543412653098634964, −11.09321119924917050466175994481, −10.61558038956082681325066183684, −9.96418410669946726243277746687, −8.93170228163670338417933952958, −8.25926011967767572891422294835, −7.27109223612465231792334633401, −6.41087111960759476287737806608, −5.46017155062815337087955502437, −4.76038011590449669430977475197, −3.94071358901383928388868145451, −2.41706989661418861482442011562, −0.94116380698261313926298721238, −0.17471679606611690837039717955, 1.25100286403648621919760714935, 1.89202763095136570914972527539, 2.71974439552276813619299080745, 4.14104203754982607239220463198, 5.06251483871418827833606190581, 6.24671638263736152773959967530, 7.05412569249403680918805147150, 7.997507424321556841817771076569, 8.63555013667460603741870686900, 9.46875827336200011550783877053, 10.70820515320942327704725082276, 11.26472896134775629328645756612, 11.97948490451426068524432027049, 12.51935589976624438382687939086, 13.54749091227742789384469207159, 14.19501602463512933432698772116, 15.57278252790842996778710071662, 16.33379240367304311209866436311, 17.38579266672959387146074882403, 17.74348395236357326716748414616, 18.40774950673305853215358942240, 19.36411809576987914106367959084, 19.58233689952991525758356266559, 20.88756200398127596459364821496, 21.50401010922067396901566743755

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